I found the Sports Park parking the best. There are fields for sports and a bike rental place right there.
Via buffalobayou.org: There are public parking areas available at Lost Lake, Eleanor Tinsley Park and The Water Works, as well as along Allen Parkway, Memorial Drive, and city streets surrounding Buffalo Bayou Park. When parking along city streets, please pay close attention to signage, be neighborly and use the pedestrian bridges to safely access the park. Also: there's parking at the Police Memorial.
the parking lot is little bit farther but is free. Please pass the park for about half a mile, you see it in your right hand on the bank of Buffalo River
We parked across the street and got subway then walked across the street to the park
Thanks! Your answer is awaiting moderation.