  We study geometry of the weak almost para-$f$-structure and its subclasses.
This allow us to produce totally geodesic foliations and also to take a fresh
look at the para-$f$-structure introduced by A.\,Bucki and A.\,Miernowski. We
demonstrate this by generalizing several known results on almost
para-$f$-manifolds. First, we express the covariant derivative of $f$ using a
new tensor on a metric weak para-$f$-structure, then we prove that on a weak
para-${\cal K}$-manifold the characteristic vector fields are Killing and $\ker
f$ defines a totally geodesic foliation. Next, we show that a para-${\cal
S}$-structure is rigid (i.e., a weak para-${\cal S}$-structure is a para-${\cal
S}$-structure), and that a metric weak para-$f$-structure with parallel tensor
$f$ reduces to a weak para-${\cal C}$-structure. We obtain corollaries for
$p=1$, i.e., for a weak almost paracontact structure.
