  This work is devoted to the study of families of infinite parabolic iterated
function systems (PIFS) on a closed interval $X\subset\mathbb{R}$ parametrized
by $\bold{t}\in U\subset \mathbb{R}^d$ with overlaps. We show that the
Hausdorff dimension and absolute continuity of ergodic projections through the
families of infinite PIFS are decided \emph{a.e.} by the growth rate of the
entropy of the sequence of concentrating measures and Lyapunov exponents of the
family of truncated PIFS, under transversality of the families essentially. We
also give an estimation on the upper bound of the Hausdorff dimension of
parameters where the corresponding ergodic projections admit certain dimension
drop. The setwise topology on the space of measures enables us to approximate
the families of the infinite systems by families of its finite truncated
sub-systems, which plays the key role throughout our work.
