Does the trem on the Parker guitars like the Fly allow for up-bends, or does it function in basically the same way as the strat type tremolo?
To answer your original question, the Parker trem can be set up for full-floating like the Strat should be, or you can set a stop screw and increase spring tension to make it capable of downward movement only, or crank the springs even more to make it more rigid for radical bends with heavier strings, but still capable of stiffer downward movement with the bar.
Various Strats
Polytone Mini Brute
Koch Studiotone XL
Quilter OD200, 101 Reverb and Mini
1958 National lap steel
Eastman El Rey 1