The authors prove several results on the Hausdorff measure of nodal sets of solutions \(u\) of the equation \(Lu=0\) in the unit ball \(B\subset \mathbb{R}^ n\). Here \(L\) belongs to a special class of elliptic or parabolic operators and \(u\) is assumed to have a certain compactness property. For example, it is proved that the \((n-1)\)-dimensional Hausdorff measure of the set \(\{x\in B_{1/2} (0)\mid u(x)=0\}\) is less than a constant \(C<\infty\), which only depends on the dimension \(n\) and several parameters.