Meesha pretty much said it all! After many decades of research done in the field of avian diets in captivity, it's been well established that feeding a seed-only diet to a pet parrot will drastically shorten the bird's life for a number of reasons, and will also cause many health issues that manifest themselves in the outward appearance of the bird. The days of going out and buying a big bag of "Cockatiel Food" or "Large Hookbill Food" that consists of a bunch of different types of seeds with small amounts of other fillers is over, or should be.
How old is your Parrotlet? And how long have you had him, his entire life, or has he lived with other families? The reason I ask is that an overgrown beak (especially in a bird as small as a Parrotlet) is very indicative of fatty liver disease. The progression of fatty liver disease can be stopped by a diet change, but he needs to see an experienced avian vet that knows how to treat the damage that is already done. It's the same situation with his feathers, this is very indicative of a bad diet and malnutrition, and can be stopped and reversed to a point, unless he is plucking and has damaged his follicles. Have you ever seen him plucking? If he is plucking it could be because he is sick and malnourished, i.e. a physical problem, or it could be because of an emotional issue such as a traumatic incident, moving and changing environments, changing owners, etc.
Again, it would be useful to know your bird's age and background, and you definitely want to do a lot of research online about a proper diet for a Parrotlet. There is ample information not only in forums like this one (the best one in my opinion) but also on many websites that actually have tons of recipes for bird mash, chops, salads, etc. The basic skeleton to a captive bird's diet is the same between breeds, and consists of a percentage of pellets, fresh veggies, fresh fruit, whole grains, nuts, and then seeds as a treat. However, keep in mind that this is only a general outline for all captive birds and each individual species has their own unique requirements. I'm not very familiar with parrotlets and have never owned one or cooked for one, so I can't help much with that.
I think the bottom line here is that you need to find an experienced avian vet, hopefully a certified avian vet, though an avian qualified vet can be very good as well depending on how much experience they have. It sounds like the vet you saw I'd not familiar with what is going on with your bird except that it's diet related. That being said, I would find a new vet asap and give them a call to set up a full intake exam with fecal smears, blood work, and possibly even x-rays. This will give your avian vet a pretty inclusive picture of what's going on and how to treat him.
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