The authors construct an approximate degree theory for a class of multivalued mappings of the form \(F= L+S\) from the domain \(D(L)\) in a real reflexive Banach space \(X\) to its dual space \(X^*\), where \(L\) is a closed densely defined maximal monotone operator and \(S\) is a multivalued map of class \((S+)\) with respect to the graph norm topology of \(D(L)\), as is indicated in the previous works. The degree theory developed makes it possible to apply continuation methods in the study of quasilinear parabolic hemivariational inequalities, i.e. multivalued initial boundary value problems of the form \[ u_t+ Au+ e= f\quad\text{in }\Omega\times (0,T);\quad u(x,0)= 0\quad\text{in }\Omega; \] \[ u(x,t)= 0\quad\text{on }\partial\Omega\times (0,T);\quad e\in\overline\partial E(u(x, t))\quad\text{in }\Omega\times (0,T), \] where \(\Omega\) is a bounded open subset in \(\mathbb{R}^N\), \(f\) is a given function defined in \(\Omega\times (0,T)\) and \(A\) is of the form \[ Au(x,t)= -\sum^N_{i=1} {\partial\over\partial x_i} a_i(x,t,u,\nabla u)+ a_0(x,t,u,\nabla u) \] and the symbol \(\overline\partial E\) designates Clarke's generalized gradient of a locally Lipschitz functional \(E\).