  In this paper, we introduce the notion of parabolic log convergent
isocrystals on smooth varieties endowed with a simple normal crossing divisor,
which is a kind of $p$-adic analogue of the notion of parabolic bundles on
smooth varieties defined by Seshadri, Maruyama-Yokogawa, Iyer-Simpson, Borne.
We prove that the equivalence between the category of $p$-adic representations
of the fundamental group and the category of unit-root convergent
$F$-isocrystals (proven by Crew) induces the equivalence between the category
of $p$-adic representations of the tame fundamental group and the category of
unit-root semisimply adjusted parabolic log convergent $F$-isocrystals. We also
prove equivalences which relate categories of log convergent isocrystals on
certain fine log algebraic stacks with some conditions and categories of
adjusted parabolic log convergent isocrystals with some conditions. We also
give an interpretation of unit-rootness in terms of the generic semistability
with slope 0. Our result can be regarded as a $p$-adic analogue of the results
of Seshadri, Mehta-Seshadri, Iyer-Simpson and Borne.
