Summary: Let \(E\) be a complex Banach space, \(c_ i\in \mathbb{C}\) \((1\leq i\leq n- 1)\), and \(A\) be a nonnegative operator in \(E\). We discuss the parabolicity of the higher-order abstract differential equations \[ u^{(n)}(t)+ \sum^{n- 1}_{i= 1} c_ i A^{k_ i} u^{(n- i)}(t)+ Au(t)= 0\leqno{(*)} \] and some perturbation cases of \((*)\). A sufficient and necessary condition for \((*)\) to be parabolic is obtained, provided \(k_ 1> k_ 2- k_ 1>\cdots> 1- k_{n-1}>0\), \(c_ i\neq 0\) \((1\leq i\leq n- 1)\). For \(A\) strictly nonnegative (Definition 1.3), \(n= 3\), \(c_ 1,c_ 2\geq 0\), a sharp criterion is given.