The following paper gives a large family of K3 surfaces for which the authors prove that the geometric Picard number is 3. The generators are quite explicit, since the surfaces are defined by $(2,2,2)$ forms on $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$.


Baragar, Arthur; van Luijk, Ronald;
K3 surfaces with Picard number three and canonical vector heights.
*Math. Comp.* **76** (2007), no. 259, 1493–1498.  MR2299785