It seems to me that all known CY3 with $h^{2,1}=1$ has the complex moduli space of the form $\mathbb{P}^1\setminus \{p_i\}_i$ for some $i \in \mathbb{N}$.

Is this true for all CY3 with $h^{2,1}=1$? Can the moduli space be, for example, a punctured elliptic curve?