I may not be able to help much. I was only in Lake Park once early last year doing the Goose Creek Loop and I didn't see any water sources at all. That was before I switched to carrying less water and I wasn't looking closely but I didn't see anything obvious. And the Trails Illustrated map, doesn't show anything obvious. You should probably plan on no water available on the Lake Park trail. Not sure what what trails you are taking to get there but If you come up from Twin Eagles on BM, there was a small trickle just north of where it connects with Hankins or Lizard Rock (not sure of what the trail is called there) in late August. Probably will be there. Then there's a small stream north of BM/Lake Park junction. That should still be available. If you turn west on BM to go to McCurdy or Bison, there's a couple small seepages that would be difficult to get water from unless you are using a filter or have a very small scoop. You can bushwack a little ways north of the trail just past McCurdy Mountain to get to McCurdy Creek which is a good flowing stream and a sure thing. If you go north on McCurdy Park, there's a creek that flows alongside for a while but I've only been there in late spring so can't vouch for that.
Sorry I couldn't be more help on this opne.