Are trails restricted for bikes?

I believe that you can ride your bike in some but others are walkers only

Yes, some trails are restricted. See: https://southcarolinaparks.com/paris-mountain/trails#jump PMSP says: We have 15 miles of shared use trails. Mountain biking is allowed every day except Saturday and the Park stays open on Tuesday evening in the Winter for those who like to mountain bike in the dark.  A printable trail map is available from the website and the Park Center has a contour model of the Park with the trail system laid out. Park personnel can also offer advice.

The trail maps show which trails are bike free.

Bikes are not allowed on Saturdays. Every other day they are allowed on most trails.