Is there any standard practice in REST to cover this case in Microservice architecture?
No, because REST does not prescribe things such as reserving certain route parameters for certain microservices. REST does not contain any definition of microservices to begin with, let alone how multiple microservices should(n't) interact or how to share/separate routes.
You're succumbing to your own arbitrary rules here. None of this is technically impossible, you're just constantly bumping into the rules that you've set out for yourself. It's time to revise the rules and see if they actually make sense or if they are unnecessary blockers.
Something has got to give here. Points 2 and 3 do touch on REST principles, so I'd argue that your question implies that you don't want to budge on these as much as you would on point 1. And in my opinion, point 1 is the weakest link:
/parent prefix is reserved for ParentService.
Why? What are you achieving by prohibiting this? Because whatever it is that drove you to accept this as a rule, it is squarely at odds with the part of your brain that offered GET /parent/{parent_id}/child as a potential solution.
The key question you need to ask yourself here is what the purpose is of "route nesting" your microservices. What are you getting out of this?
Or, another way to phrase the same quandary: is the (sub)route structure owned by an individual microservice, or is the whole route structure owned by your ecosystem (i.e. collection of microservices?
I'm not telling you which answer is correct, but you need to pick a lane here and stick with it.
Either you agree that services owning their own route prefixes is necessary, and therefore you sign yourself up for your child service needing to have a route that is distinct from the parent route prefix; or you agree that it is not necessary and therefore allow yourself to have the "get child by parent ID" endpoint make itself indistinguishable (to an outside consumer) from actual parent-related endpoints.