I agree, a Certified Avian Vet or Avian Specialist Vet shouldn't have any issue with being bitten at all by a parrot at all...Was this vet an "Exotics" Vet? If so, then you're probably better-off not having them do it anyway...
As Sailboat already mentioned, if this is an issue with your bird's beak actually growing too-long and really needing to be filed-down once a year, then you really should have routine, Baseline blood-work done, because if your bird's beak is actually growing long-enough that it needs to be filed-down each year, then this is an indication that he may be suffering from Fatty Liver Disease, or other liver issues. Your bird's beak shouldn't ever need filed-down at all because it's too long. If you provide a cement-perch for both his toenails and filing his beak, along with the Mineral Block and other toys, then his beak shouldn't ever grow too long to the point of needing to be "shortened" in any way...If it's simply a matter of his beak flaking or looking rough, then that's an issue that you should be able to treat at home...But usually when you have to take to a Vet to be filed or God forbid they use a power rotary-tool like a Dremel, which is quite dangerous and can actually prove fatal for the bird for a number of different reasons, there is a health/nutritional issue going on and blood-work should be taken...
Otherwise, I would buy some cement perches and place them throughout his cage, along with different Mineral Blocks and toys that encourage him to file his own beak on them. Then it shouldn't be necessary at all, except maybe a simple and quick hand-filing at home to shape the point at the end of his beak, but that's it, no length should need to be taken-off...