Sunday Morning coming down!

Since $7^2 \equiv 49 \equiv -3 \pmod{13}$ and $5^2 \equiv 25 \equiv -1 \pmod{13}$ , we see that $7^{2a} - 5^{2a} \; \equiv \; (-3)^a - (-1)^a \; \equiv \; (-1)^a(3^a-1) \pmod{13}$ and so we are interested in integers $a$ such that $3^a \equiv 1 \pmod {13}$ . Since $3^2 \equiv 9 \pmod{13}$ and $3^3 \equiv 27 \equiv 1 \pmod{13}$ , we deduce that $3^a \equiv 1 \pmod{13}$ precisely when $3|a$ . Thus there are $\boxed{333}$ integers $a$ between $1$ and $1000$ with the required property.