For the solution, I think 10C2 is over counting the parallelograms. I don't see how you can make a parallelogram with the following parallel lines.
I think the answer is either 3*(11C4) or 3*(12C4).
I solved it two ways. One is where you choose 4 points on the bottom and extending them should give you a unique parallelogram, but I'm not sure. So I think it should be 3*11C4?
The other way is like "case-work." You let the bottom base be a certain line and do case work on the "height." And then you also do case work on the different bases. So for each orientation of the parallelogram, it should be (10C2 + 9C2 + 8C2 +... )+ (9C2 + 8C2 +...) + (8C2 +...) +... (3C2+2C2) + (2C2) which is 11C3 + 10C3 +... = 12C4. So the total is 3*12C4.