I’m not much of a wah guy…but I own two. 
I had an idea for an original composition that- as I heard it in my mind- absolutely required a wah.  So I set about shopping for one.  Ultimately, I settled on a T.Rex Gull wah, for a variety of reasons.  Problem was, by the time I found the product, it was discontinued.  I found 2 websites selling their final ones.  I ordered one…but couldn’t tell if my order went through.  I tried again, but it was gone.  Their website had automatically updated their inventory, but I STILL couldn’t tell if I had been the purchaser or some else.  So I ordered the one from the other website.
They both arrived on the same day.
Thing is,  I’ve never even taken either of them out of the box!

 So unfortunately, I can’t say if it’s parkable.
And the cherry on top: a short time after retirement of The Gull, T.Rex released the Shafter wah, which, AFAICT, is the identical pedal but in a more traditional appearing housing*.
All that said, one of my axe-slinging friends has a Fulltone Clyde that he loves, and one of his main methods for using it is in a parked position.
* just saw a comment on another thread in which someone at T.Rex said the Shafter is based on the Gull, but isn’t identical.  Didn’t illuminate anything further.