Parasemifields (i.e., commutative semirings whose multiplicative semigroups are groups) are considered in more detail. We show that if a parasemifield $S$ contains ${\mathbb{Q}}^{+}$ as a subparasemifield and is generated by ${\mathbb{Q}}^{+}\cup \left\{a\right\}$, $a\in S$, as a semiring, then $S$ is (as a semiring) not finitely generated.