No such integer pair exists. This can be proved using the MOD method.

It can be shown that for any cube modulo $7$ can only result in $x^3\equiv 0, 1, 6\pmod{7}$. Meanwhile, any power of $2$ modulo $7$ can only result in $2^y\equiv 1, 2, 4\pmod{7}$. Hence $2^y+15\equiv 2, 3, 5\pmod{7}$. Therefore, the two sides cannot be equal.