We were there in April, campermon was there is June, both reports with pictures are on this website for example
Palo Duro Canyon SP (4/2008) pictures are linked in first post
The hike to the lighthouse formation is great but very dry, leave real early if you go. lots of sunscreen, hat and water.
Campermom has June pictures from Palo Duro under her post of other parts of the park
I think we had a discussion on those hike in sites at Palo Duro before, you really do not need that for quiet but here is what the park book says
"Not shown on the park map is a primative hike in area that is 1/2 to 3/4 miles, ground fires are not permitted with potable water at the trailhead and restooms 1/4 mile from the parking area."
We did not go looking for them so I can't say much else, the 1/2 mile from water deters most folks as carrying water is not fun and no ground fires means a stove.
I noted above in some areas like Hackberry you will be almost be alone anyway. Just remember on the trails that park has rattlesnakes