Thank you for your interest in OpenStreetMap and for creating an account so we can follow up with your comments.  I'm from Aiken and try to address new notes in SC where I can.  I'll try to break your description into some specific map issue:

1) Regarding the park name, it would be the practice of OpenStreetMap to "map what's on the ground".  If there signage at the park that says "Hazel Parker Playground", that's what it should be on the map.  Is this correct?  I can assign "Hazel Park" as an alternate name if that's what you hear people calling it.

2) Regarding dog park, it is currently marked as having extents bounded by a 5 ft fence as shown in the link below.  If this is inaccurate, I'd suggest you mark specifically, with an additional note or notes, where the fence should be move.  I've reviewed two sources of commercial imagery and don't think I can improve this without information from on-the-ground.
https://www.openstreetmap.org/way/940446336

3) A fair number of trash cans and benches are on the map.  Feel free to note the locations of additional park features individually, and we can add them.

4) Is this the place you mention parking?  It looks properly marked as parking.  Is any change needed?
https://www.openstreetmap.org/relation/19609315