Finitely generated and parafree not implies free

Statement
It is possible to have a finitely generated parafree group that is not a finitely generated free group (and hence, not a free group).

In fact, the example we construct will be a finitely presented group.

Proof
Suppose $$i,j$$ are integers such that ij \ne 0 (i.e., neither of them is zero). Consider the group:

$$H_{i,j} = \langle a,b,c \mid a = [c^i,a][c^j,b] \rangle$$

The group $$H_{i,j}$$ is parafree. Explicitly, it has the same lower central series quotient groups as free group:F2.