  We introduce a new a family of $Z(N)$ multispins quantum chains with a
free-fermionic ($N=2$) or free-parafermionic ($N>2$) eigenspectrum. The models
have $(p+1)$ interacting spins ($p=1,2,\dots$), being Hermitian in the $Z(2)$
(Ising) case and non-Hermitian for $N>2$. We construct a set of mutually
commuting charges that allows us to derive the eigenenergies in terms of the
roots of polynomials generated by a recurrence relation of order $(p+1)$. In
the critical limit we identify these polynomials with certain hypergeometric
polynomials ${}_{p+1}F_p$. Also in the critical regime, we calculate the ground
state energy in the bulk limit and verify that they are given in terms of the
Lauricella hypergeometric series. The models with special couplings are
self-dual and at the self-dual point show a critical behavior with dynamical
critical exponent $z_c=\frac{p+1}{N}$.
