  Given a family of local systems on a punctured Riemann sphere, with moving
singularities, its first parabolic cohomology is a local system on the base
space. We study this situation from different points of view. For instance, we
derive universal formulas for the monodromy of the resulting local system. We
use a particular example of our construction to prove that the simple groups
$\PSL_2(p^2)$ admit regular realizations over the field $\QQ(t)$ for primes
$p\not\equiv 1,4,16\mod{21}$. Finally, we compute the monodromy of the
Euler-Picard equation, reproving a classical result of Picard.
