So I have a set P={ $p(\alpha,\beta,\gamma)=\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$ $|$ $\alpha,\beta,\gamma$ $\in R$}. I needed to show P is a Lie group, which I have done.
I need to parametrise P, and I was asked to show that it is $R^3$, and, to show that the group multiplication is:
$p(\alpha,\beta,\gamma).p(x,y,z) = p(\alpha+x,\beta+\alpha z + y, \gamma+z)$
So I multiplied the matrices $\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$. $\pmatrix{1&x&y\\0&1&z\\0&0&1}$ = $\pmatrix{1&\alpha+x&\beta+\alpha z + y\\0&1&\gamma+z\\0&0&1}$.
My question is, how would I parametrise the group above?
You've already parametrized the group. The way you've written it establishes a bijection between the elements of $\mathbb R^3$ and the elements of $P$; that's what a parametrization is. The matrix multiplication you've carried out verifies the given multiplication law for the parameters, because the product of the matrices parametrized by the triples on the left-hand side is indeed the matrix parametrized by the triple on the right-hand side. There's nothing more to it than that.