Well, I can speak from experience that the "Fly Jars" or "Fly Trap Bottles" work wonders...I kept the bottle out in my kitchen for a week (it says they are good for 30 days before having to dump them and insert a refill-pack), and every fly in my house is gone. And there were tons of them...they were fruit-flies, but it says they are good for all flies. I mean every little fly is gone, I haven't seen a single fly in the couple of weeks since I threw it away...And they are totally bird-safe, more-so than any spray is going to be...
I will point out though that there is no way you could keep one of these jars/bottles out for 30 days, as after a week it started to smell (dozens of dead flies, yeah, it stunk)...I noticed that the actual stuff inside the jar (you add a packet of powder and then fill the far with water) turned into a hard gelatin kind of substance, and the flies are attracted to it, crawl into the jar/bottle, and then get stuck in the gelatin and die...and the nice thing is that they can't leak if you knock them over because the liquid turns into gelatin...But it did stink pretty badly after a week, though all my flies were totally inside the bottle and gone within 2-3 days...
$6 at Home Depot...They also sell little organic fly-jars, same thing but organic and made by a small company, online...They come in different scents that are supposed to cover the smell of the gelatin...haven't tried these, only the "Fly-Trap Bottle" from Home Depot. Don't need any more, one did the trick.
****Just a note to the member who posted above about the essential-oil spray, you must investigate each and every individual essential-oil that you spray/burn/diffuse in your home for any reason, as some are fine, some are extremely toxic to birds (like Tea Tree Oil diffused), some will make them very sick...Cinnamon, Thyme, Sage, Oregano, Bark, Tea Tree, Pine, Basil, Wormwood, Spruce, Birch, Juniper, Marigold, Pennyroyal, Ylang Ylang are the most toxic to diffuse/spray into the air for birds, Tea Tree being the most toxic, followed by the "tree/wood/bark" scents...Rosemary as far as I know is fine, but the Thyme oil was not good, hopefully your bird didn't breath a lot, or any of it in, just watch him for any signs of illness, and get him to his CAV asap if you notice any, specifically any breathing/respiratory signs/symptoms...And "hot" essential oils are usually toxic for birds, for example Cinnamon, Oregano, Sage, Thyme, Nutmeg, Clove, etc., as they cause irritation throughout the respiratory system....
It sounds like the insect spray your parents used were supposed to be "natural" or similar, as it consists of essential oils and then the phenethyl-proprionate, which is also made from certain natural plant-esters...However, "natural" doesn't mean "bird-safe", and again not all essential oils are safe, some are deadly to birds. And most-all "cheap" essential oils are not safe for birds, regardless of which they are, as most "cheap" essential oils have additives, or even worse, are not actually natural, essential oils at all, they are synthetic scents made in a lab, the equivalent to burning a scented candle unfortunately...Real, natural, organic Essential Oils that are unadulterated and have no additives at all are extremely expensive....
On a side-note, I was buying vitamins at Walmart a couple of weeks ago and I noticed that Walmart started selling a new brand/type of Essential Oils...They always sold very cheap ones along with cheap diffusers in the vitamin/supplement aisles, but this is different...There is a huge case taking up the entire top shelf of one of the vitamin aisles, it's wooden, and it's some new brand of 100% natural, organic Essential Oils, all in this huge, wooden case...They probably have 30-40 different oils...I read the bottles and they are in-fact pure, natural, organic, no additives, etc...and yep, even at Walmart they cost a small fortune for a tiny little bottle with a few ounces in them..I'll stick to buying mine at Wegmans, same price and better quality...