As for the reason why there are two parameters, the reason is as follows.

Pythagorean triples are (or may be) organized into m groups, each with n triples. As such, m and n simply designate placeholders for any given triple. Within each group, then, only one parameter is required; it is just that there are an infinite number (m) of groups.

The first few groups look like this:
----m = 1----m = 2 -----m = 3 ----m = 4------
n=1<2, 0, 2><8, 0, 8><18, 0, 18><32, 0, 32>
n=2<4, 3, 5><12, 5, 13><24, 7, 25><40, 9, 41>
n=3<6, 8, 10><16, 12, 20><30, 16, 34><48, 20, 52>
n=4<8, 15, 17><20, 21, 29><36, 27, 45><56, 33, 65>

You may notice that the first term in each group is 2m^2, and that this is also the difference of c - b.

Meanwhile, the first term, a, in each triple is 2m(n+1).

The second term, b, is a little more complicated. In each case, it is simply the sum of odd numbers, but the sum leaves off some odd numbers (the first non-zero term in each group shows you where the odd numbers "start" from). You might note that in m=1, b=n^2-1. However, this is simply fortuitous, as the other groups do not reflect this pattern.

Rather, for m = 1, b = n^2 - 1
for m = 2, b = (n+1)^2 - 4
for m = 3, b = (n+2)^ - 9

or, in general: b = (n + m - 1)^2 - m^2

Lastly, c = b + 2m^2

So, this is why there are two parameters. If you decide just to look at, say, m = 2, then you get:

a = 4(n+1)
b = (n+1)^2 - 4
c = (n+1)^2 + 4

n=1 --> a = 8, b = 0, c = 8
n=2 --> a = 12, b = 5, c = 13

etc

Hope this helps.