This is pseudo-science and a common misconception. There is no genetic reason or support for mandatory detrimental tradeoffs.
Historic breeding programs often allow side effects as a practical matter, but that doesn't mean they are detrimental or required.
Sometimes genetic difference is all upside just like a mutation can be all downside.
We might be willing to accept the side effects but that doesn't mean they won't inevitably occur.
With respect to wild parrots, it's not obvious that improvement is possible, let alone has disadvantage. Assuming it is possible, there are numerous reasons why it might not already exist. No advantage, low negative advantage, negative incremental advantage, and low probability are all possible reasons. Not every viable state has a viable incremental path to it.
Most genotypes or "random mutations" are totally neutral for an organism, but let's say that of the notable ones, it's 50/50 whether they will be "good" or "bad".
If we find the "good" smart parrot, it's likely they have some other "bad" stuff we can't see or just don't care about. When you keep inbreeding, that other bad stuff will be fixed along with the intelligence. This is purely a limitation of our knowledge of the genome and lack of other inputs. You can out-cross to reduce the bad traits, but you risk losing some of the intelligence too. Theoretically there is another bird out there that solves the "bad" trait without affecting intelligence, we just don't know which it is.
Nothing stops you from having a win-win-win. You could randomly find a bird that is super smart with rainbow feathers and the voice of an angel... It's just incredibly unlikely to roll all those attributes in one individual and keep them through multiple generations.
As to why wild populations haven't naturally evolved to be smarter, that is an entirely separate discussion about countless competing selection pressures and how genes spread through larger populations.
That isn't always true. We know in humans there are genes that depending on which variation you have, you will either be more social or have better memory. According to AI I'm thinking of BDNF (Brain-Derived Neurotrophic Factor), but digging deeper doesn't indicate that is correct. Either way there often are trade offs in genetics.
Though of course genetics is only one factor. How your are raised/taught also makes a difference.
Having trisomy 21 or exons in your myosin genes don't give you superpowers in another part of your life. There are people with excellent memory and social skills, and people that lack both.
By way of analogy, nobody would say that all possible airplane designs are equal and there are only trade-offs. Some are superior in every possible respect to others.
The appropriate response unknowns is not to declare that tradeoffs are always the case, let alone equal