Your puzzle has 7790 solutions (and their mirror images). I used my own polyform solver (here), which took about 11 minutes. It is slow because it is rather flexible, but you can use faster solvers that are more specific for polycubes, such as the Polycube Puzzle Solver already mentioned in the comments above.
I doubt that there is any 4x4x4 puzzle with a unique solution when you use pieces as small as pentacubes. You are likely going to need one or more larger pieces, or some other way to restrict the placement of the pieces. You could use a colouring scheme (e.g. checkerboard colouring).
I'll do a bit of playing around with my solver to see if I can find something interesting.
I decided to use 10 hexacubes instead of 12 pentacubes, to make it easier to find sets of pieces with few solutions. My best effort is the following set.
The 10 flat hexacubes consisting of a 1x1x4 spine with two cubes attached to the spine:
A B C D EE
AAAA BBBB CCCC DDDD EEEE
A B C D
F F G G HH I J
FFFF GGGG HHHH IIII JJJJ
Plus the L tetracube:
This has only 3 solutions (and their mirror images).
In the comments below, theonetruepath writes:
Pack a 2x3x4 with any 12 pentacubes and any one tetracube, in all ways. Include all symmetric packings. Count how many times each piece occurs. Make a list of the 12 pentacubes that occur least often plus the tetracube that occurs least often. This set of pieces does not pack the 4x4x4.
Unfortunately I was unable to reproduce this. The set of pentacubes I get still has many solutions when combined with any tetracube. I therefore still stand by my view that pentacube pieces are too small to produce a puzzle with a unique solution.