An abelian group $G$ is quasipure projective if for every pure subgroup $P$ of $G$, the canonical map $G\rightarrow G/P$ induces an epimorphism $\text{Hom}(G,G)\rightarrow\text{Hom}(G,G/P)$. Dual definition applies to quasipure injectivity. The authors characterize these groups $G$ completely in the torsion case [cf. K. Benabdallah and R. Bradley, Canad. J. Math. {\bf 29} (1977), no. 1, 107--110; MR0427495 (55 \#527); Benabdallah and A. Laroche, Ann. Sci. Math. Québec {\bf 1} (1977), no. 1, 63--65]. It is shown that a $p$-group $G$ is quasi-pure projective if and only if it is either a direct sum of cyclic $p$-groups (i.e., pure-projective) or a direct sum of a bounded $p$-group and a divisible $p$-group. $G$ is quasi-pure injective if and only if it is a direct sum of a divisible $p$-group and a torsion-complete $p$-group (i.e., pure-injective in the class of $p$-groups).