I don't know why your CAV would tell you to "cut back on fresh foods", at least as far as the fresh veggies and greens, as they are not an issue at all, considering that most parrots don't eat the whole batch of chop when you give it to them...As far as fresh Fruit, a lot of people do feed their birds and their reptiles too much (they typically think of fresh veggies and fresh fruit as one in the same), as all fruit is loaded with sugar, and that sugar turns into fat and is stored in their livers, especially in captive/pet birds that aren't flying miles and miles every day...So I do fresh Fruit only 2-3 days a week, but as far as fresh veggies and fresh, dark, leafy greens, they can have as much as they'll eat every day, because they don't each much to begin with...
***There are A LOT of different veggies and dark, leafy greens that do 2 different things that people blow-out of proportion, considering the small amount of fresh veggies and dark, leafy greens that birds/reptiles usually eat in a day (this same topic comes up ALL THE TIME, CONSTANTLY in the Bearded Dragon forums, like 20 times a day, because it's more crucial to Beardies because fresh veggies and dark, leafy greens make-up at least HALF of an adult Bearded Dragon's daily diet, and usually it's actually more than half, because after they hit 2 years-old and are done growing, that's when they naturally stop eating much live-insect protein because their bodies don't need it anymore and they just end-up getting obese, plus they get most-all of their hydration from fresh veggies and greens; my almost 3 year-old female Beardie cut herself down and now only eats live-insect protein once or twice a week, so it's ALL salad):
#1) Some veggies bind Iodine, as already mentioned, and the worry is that the animal will become Iodine-deficient (we get iodine in everything we eat because of iodized salt, but obviously they don't eat a lot of food with salt)...However, if your parrot is eating a pellet or a healthy, varied seed-mix as their daily staple, then they are getting more than enough Iodine, so the small amount of Broccoli and other Cruciferous and Goitrogenic veggies/greens that your parrot eats in a day isn't going to bind enough Iodine to cause a reduction in Thyroid Hormones. They'd have to basically be eating nothing else but Crucifers/Goitrogenic veggies/greens every day, all day long for them to become Iodine-deficient...
****And just to make a point here, Broccoli is only one of the MANY Cruciferous/Goitrogenic veggies/greens that bind Iodine. So if this was really an issue and harmful to our parrot's health and was going to cause an issue, we would be hearing about parrots suffering from Thyroid issues all the time, constantly. As of right now, I don't think that I've ever heard of a parrot suffering from an Iodine-deficiency or having a Thyroid issue or being deficient in Thyroid hormones. ****Here's just a few of the Cruciferous/Goitrogenic veggies/greens that bind Iodine:
-Broccoli
-Brussels Sprouts
-Cauliflower
-Kohlrabi
-Radishes
-All types of Cabbage
-Bok Choi and Pak Choi
-Arugula/Rocket
-Collard Greens
-Mustard Greens
-Turnip Greens
-Kale
-****SOY****
So you can see how many fresh veggies/greens that most people feed their parrots, reptiles, rabbits, guinea pigs, etc. every single day bind Iodine. If it was an issue, we would be hearing about lots of parrots suffering from issue due to Iodine-deficiency...Especially with the Soy; how many parrot pellet brands contain Soy?
***The other issue with pretty much the same group of fresh veggies, though this particular issue is mainly caused by ALL of the dark, leafy greens and not so much by the Broccoli, Cauliflower, Soy, etc., is that the all contain a very high content of Oxalates, and Oxalates bind Calcium. This is a HUGE topic of conversation and controversy in the Bearded Dragon community because daily Calcium intake is so crucial to a Beardie's survival, so much so that we have to dust their live insects in both a Multi-Vitamin powder and a Calcium powder that usually contains Vitamin D3 as well, again to encourage the binding of as much Calcium as possible from the food that they eat.
The Oxalate issue is a very frustrating one, because the very same dark, leafy greens that contain the highest Oxalate content are also the ones that contain the highest Calcium content. The general consensus for Bearded Dragons and other reptiles that are so dependent on a large, daily Calcium intake and absorption is that it's much better to continue to feed adult Dragons the dark, leafy greens every single day, get them not only their large intake of Calcium and other Minerals, Vitamins, etc., but also their water intake, and simply make sure to dust all of their live-insects in Calcium Powder w/D3, and on days when they are not eating live-insects you should sprinkle the Calcium w/D3 on their salads.
And although Calcium is extremely vital to every living creature, including humans, because every single cellular process/function requires Calcium to occur, for parrots it's the same situation as the binding of Iodine by the very same group of veggies/greens: They are getting ample Calcium from their daily staple of either pellets or a healthy, varied seed-mix, and also from their fresh veggies and dark, leafy greens, so the Oxalate content in these veggies and dark, leafy greens is not going to bind enough of the free-Calcium they are ingesting in everything they eat to cause a deficiency.
HOWEVER, this does put the importance of Cuttlebones and/or Mineral Blocks into perspective, specifically with all female birds who are laying eggs. While people always try to get their egg-laying female birds, or female birds who have just laid a clutch of eggs and are feeding a bunch of chicks directly after laying the eggs, to eat as many fresh veggies and dark, leafy greens as they are willing to eat, simply because of their high Calcium content (which is the correct thing to do in my opinion), in this specific situation with laying/feeding female birds, the extra-high Oxalate content in these very same veggies and greens could actually have a negative effect...SO...this is exactly why I always have multiple Cuttlebones and Mineral Blocks all over the cages/aviary where my breeding-pairs lived and raised their chicks, and I also fed the females Egg-Food every single day along with their pellets/seed-mix.
***The veggies/greens with the highest Oxalate content:
-Spinach*
-Kale*
-Swiss Chard
-Arugula
-Bok Choi and Pak Choi
-Collard Greens
-Mustard Greens
-Turnip Greens
-Dandelion Greens
-Endive
-Escarole
-Romaine and other dark green lettuces
-Sweet Potatoes
-Beets
-Rhubarb
-*****PEANUTS****** (Yet another reason to avoid feeding your birds Peanuts, lol)
-****Most Tropical Fruits that are native to Southeast Asia/Indonesia or Central America/Mexico, such as Star Fruit (has the highest Oxalic Acid content of any fruit), Mango, Papaya, Guava, etc.
-Grapfruit (also high in Vitamin C, which greatly enhances the absorption of Iron, so Grapefruit is just not a good fruit for birds all the way around)
*Spinach and Kale contain the highest Oxalic Acid content of any food; they also have the highest Calcium content of any veggie/green. As such, they are a hot-topic in the reptile world, especially the Bearded Dragon world...It's been widely accepted that Bearded Dragons should NEVER be fed either Spinach or Kale of any variety due to their extremely high Oxalate content...I have never fed my Bearded Dragon Spinach, and I also don't feed it to my birds for the same reason. I myself have had chronic Kidney Stones starting when I was 19; I've had surgery to remove them, and I've probably passed somewhere around 100 stones that I have been aware of passing, meaning they caused me a problem such as PAIN!!!!!!!!!!!!!!!, blood in my urine, bacterial infections in either my Urinary Tract and a few times in my Kidneys, etc. (all of my stones have always been Calcium Oxalte stones)...So there is NO SPINACH OR KALE in this house anyway.
Having Spinach or Kale in my house would be the equivalent to a Vampire cooking with Garlic in their home...It is my Kryptonite, along with Black Pepper and unfortunately Tea (both extremely high in Oxalate content)...
SOOOOOOO.....Bottom line after me typing all of that out is basically when it comes to your parrots, as long as you feed them a regular, daily diet that is varied, low in fat and high in protein, vitamins, minerals, and amino acids, and as long as you always provide a Mineral Block for them, then the small amount of these veggies and dark, leafy Greens that they eat isn't going to hurt them in the least, and their benefits to your parrots are endless...