"The program is functional, fast, and finds a solution..."

Solution 1:

I think the completely-parallel construction would be

The demonstrations show that Program A is functional, is fast, and finds a solution that Program B misses.

However, I don't think that flows any better than the original.

Of the variations you list, my second choice would be the one that puts another and between "functional" and "fast":

The demonstrations show that Program A is functional and fast, and finds a solution that Program B misses.

The capable of finding version is grammatical, but you're right, it means something slightly different than finds. I wouldn't suggest the functions variation, because I think it's even worse in terms of parallelism than the original.

Edit: on re-reading, I think the double-and version needs another word:

The demonstrations show that Program A is functional and fast, and it finds a solution that Program B misses.

Solution 2:

Here is another wording that fixes the problem and (I think) sounds less awkward than "functional and fast and finds...":

The demonstrations show that Program A is not only functional and fast, but also finds a solution that Program B misses.

Solution 3:

You can also split it in two sentences:

The demonstrations show that program A is functional and fast. [Moreover,] They [also] show that program A finds a solution that program B misses.