A non-polynomial reduction
No. As a counterexample pick any non-trivial problem $P_2$, i.e., a problem with at least one yes instance $I_{\text{yes}}$ and at least one no instance $I_{\text{no}}$.

To reduce an instance $I_1$ of $P_1$ to an instance $I_2$ of $P_2$ first solve $I_1$ (e.g., by brute force). If the answer to $I_1$ is yes, then let $I_2 = I_{\text{yes}}$, otherwise $I_2 = I_{\text{no}}$.