Is there a way to create a default parameter list so that every time I create a new schematic part, I get the same list of parameters?
I don't think this is exactly what you want but I usually set up a few part templates in the schematic library. I have a one for resistors, capacitors, ICs, etc that have various parameter names and placeholders for values. Then to make a new part I just copy and paste the template parts and then edit them. Since vault parts don't have the parameters I want, I sometimes copy and paste the symbols graphics from them into the newly pasted template part, though this doesn't seem to allow me to draw a window-box and then copy-paste.
To copy the schematic library parts just right click, copy and the paste or use Ctrl-C and Ctrl-V. The pasted part will get a _1 added to the name (symbol reference). For whatever reason, using the menu bar (Home > Copy) doesn't seem to work for me.
Also, I have found that when in the schematic library panel with the component properties open, I can right click and then select all. Then press Ctrl-C. This should copy all the parameters to the clipboard. After this I can paste into a text editor or even into MS Excel as text.
Alas, you cannot then do the reverse and copy an entire parameter set into a component. But you can at least do the string one at a time. It helps save a little time for me.