The differential equation \(u''+Bu'+Au=0\) in a Hilbert space \(X\) is considered. Conditions entailing that this equation generates a corresponding equation in a larger extrapolation space which is parabolic are studied. For example, if \(A\), \(B\) are two positive, selfadjoint operators in \(X\), and \(D(A)=D(B^{1/\alpha})\), where \(1/2<\alpha\leq 2/3\), then the pencil \(z^ 2+z\tilde B+\tilde A\) is parabolic in the extrapolation space \(Y=D(B^{1/2})^*\), \(D(B^{1/2})^*\) denoting the adjoint space of \(D(B^{1/2})\) and \(\tilde A\), \(\tilde B\) being the closures of \(A\), \(B\) respectively, as operators in \(Y\). Some relations between solutions of the original equation and extrapolated equation are proved. An example of applications to initial boundary value problems for partial differential equations is given.