I use a Wilton cookie press and have a couple of suggestions. I always use the recipe that came with the cookie press. It does call for butter not shortening. Could this be be part of the problem you are experiencing?
While I’m waiting for my sheet of cookies to bake, I put the bowl of dough in the refrigerator. It keeps it chilled enough that the dough is works better for me in the press. When the dough gets too warm it won’t release from the press easily.
My press came with a number of discs for different shapes. I’ve only found a couple that work easily and with a proper release of the dough. Usually though, the first cookie of each filled tube is a mess and I just throw it back in the bowl. After that it usually takes two clicks while holding the press firmly against the cookie sheet.
I don’t use parchment paper. It’s hard enough to keep the press firm against the cookie sheet. It seems that the parchment paper would move causing problems with the cookie shape.
i hope this helps you before you send the cookie presses to the land fill!