Can you put that you worked in your parent's restaurant on a resume?

Of course you can put this on your resume. Assuming you actually worked (rather than hung out), this is legitimate work experience.

The amount of weight this will carry will depend on what you did, how long you did it, etc.

If the name of the restaurant is Carries Mom and Dad's Restaurant and you are Carrie then it might need a little splainin' but it really doesn't matter who owns the restaurant. You have experience doing what ever you did there whether it was washing dishes, waiting tables or sweeping floors. It was a real job, you learned some skills, those skills are marketable. Go for it.

It's job experience, so yes - you can put it down. Unless the restaurant has your surname as part of the restaurant name, it's not going to be obvious either. Carries as much weight as any other job.

Yes absolutely... I would hire a person that worked in a family restaurant over others because it shows a sence of family in the workplace.