I find this quite relatable, it sounds like an anxiety thing.
In my own reflection, I'm convinced this comes from spending all our developmental years experiencing social reactions/consequences we couldn't predict, from NT peers and authority figures.
The parents are also more anxiety-inducing because if they get it in their heads to get upset, for whatever reason, they have the power as customers to make life awkward, if not difficult, for you.
Assuming you can mask when you need to, I suggest treating it as a professional skill, practice greetings, scripts etc. Try to reduce the reliance on on-the-spot cognition.
If you do this, it means really putting yourself out there for the sake of the job, and deserves to be recognized as such and compensated with self-care on other occasions.
I hope it goes well.