  We present a general study of the large family of exact integrable quantum
chains with multispin interactions introduced recently in \cite{AP2020}. The
exact integrability follows from the algebraic properties of the energy density
operators defining the quantum chains. The Hamiltonians are characterized by a
parameter $p=1,2,\dots$ related to the number of interacting spins in the
multispin interaction. In the general case the quantum spins are of infinite
dimension. In special cases, characterized by the parameter $N=2,3,\ldots$, the
quantum chains describe the dynamics of $Z(N)$ quantum spin chains. The
simplest case $p=1$ corresponds to the free fermionic quantum Ising chain
($N=2$) or the $Z(N)$ free parafermionic quantum chain. The eigenenergies of
the quantum chains are given in terms of the roots of special polynomials, and
for general values of $p$ the quantum chains are characterized by a free
fermionic ($N=2$) or free parafermionic ($N>2$) eigenspectrum. The models have
a special critical point when all coupling constants are equal. At this point
the ground-state energy is exactly calculated in the bulk limit, and our
analytical and numerical analyses indicate that the models belong to
universality classes of critical behavior with dynamical critical exponent $z =
(p+1)/N$ and specific-heat exponent $\alpha = \max\{0,1-(p+1)/N\}$.
