In Section 9 of the paper
I. Shimada: An algorithm to compute automorphism groups of (K3) surfaces and an application to singular (K3) surfaces, Int. Math. Res. Not. 2015, No. 22, 11961-12014 (2015) ZBL1333.14034
there are many examples of complex elliptic K3 surfaces with Picard rank 3 and having (infinite) automorphism group containing involutions (in fact, $\operatorname{Aut}(X)$ contains a copy of $\mathbb{Z}/2 \ast \mathbb{Z}/2$).