I'm sure there's no reason why you couldn't park in the parking lot at WRP. You just have to ride on city streets for a block or two to get on the trail. Or you could park in the shopping center on Arcola (can't remember what's there, maybe a Strosniders or Magruders? Regina would know) and pick up the trail from there.

The Sligo trail meets up with Northwest Branch/Northeast Branch/paint branch trails that Owlice mentioned. The trail goes over to Lake Artemesia in College Park (and maybe further, but that's as far as I've gone.) If I read those directions in your post right, you'd take a right when you reach the T after you cross the last bridge, and that would put you on the Northwest Branch trail (taking a left you'd be on the NW branch trail, but it ends after about a mile). There are some turns that get a bit tricky, but generally it's pretty obvious where you need to go.