Eye lid : a light approach

Hi everyone,
Rather new in character rigging. I need a cartoon like character. I use to make eye lids with a half sphere, and to rotate in on the x axis to open/close the eye.
I’ve just bought Auto Rig Pro, dans I wonder how to integrate my primitive lids.
Each has to stay centered on its eye, but must be able to rotate on given angle.
I tried to parent the lid to the eye, but when the character moves, the lid doesn’t …
Can I parent it the the eye bone, but how can I control the rotation in a chosen x like angle ?

I know, I could use a more conventional method and model an elastic lid connected to the face mesh. But I have never done this before, and it looks tricky.
T H A N K S for any help,