  We introduce a new class of parametricization structure-preserving
partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with
holonomic constraints. When the scalar parameter $\alpha=0$, the methods are
reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK
schemes based on Lobatto IIIA-IIIB pairs, which can preserve all the quadratic
invariants and the constraints. When $\alpha\neq 0$, the methods are also shown
to preserve all the quadratic invariants and the constraints manifold exactly.
At the same time, for any given consistent initial values $(p_{0}, q_0)$ and
small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0,
q_0)$ such that the Hamiltonian energy can also be exactly preserved at each
step. We provide a new variational formulation for symplectic PRK schemes and
use it to prove that the parametrized PRK methods can preserve the quadratic
invariants for Hamiltonian systems subject to holonomic constraints. The
parametric $\alpha$-PRK methods are shown to have the same convergence rate as
the usual PRK methods and perform very well in various numerical experiments.
