Here's a necessary condition. Write the symmetric polynomial $(x+y)^k - x^k - y^k$ as a polynomial in the elementary symmetric polynomials $x+y$ and $xy$, say
$$ (x+y)^k - x^k - y^k = F_k(x+y,xy). $$
Then a necessary condition for $A\in\operatorname{SL}_2(\mathbb Z)$ to be a $k$th power in $\operatorname{SL}_2(\mathbb Z)$ is that the following two conditions hold:
 1. $\det A$ is the $k$'th power of an integer, say $\det A=D^k$.
 2. The polynomial 
$$T^k - F_k(T,D) + \operatorname{Trace}(A) $$
has a root in $\mathbb Z$.

The proof is easy enough, since if $B^k=A$, then $\operatorname{Trace}(B)$ is an integer root of the polynomial.