Consider a dynamical system \((X, T)\) and a potential function \(\varphi: X\to \mathbb R\). For \(x\in X\), the Birkhoff average of \(\varphi\) at \(x\) is defined by \[ A(\varphi, x):=\lim\limits_{n\to +\infty} \sum_{i=0}^{n-1}\varphi(T^i(x)) \] whenever the limit exists. The Birkhoff spectrum allows one to study the entropy and the Hausdorff dimension of the level sets \(L_{a}:=\{x\in X:\ A(\varphi, x)=a\}\) of the Birkhoff average. Many interesting results for various dynamical systems have been obtained in the literature by using the Birkhoff spectrum, and many more systems still need to be studied, especially for those ones with nonuniformly hyperbolic behavior. In this direction, the present authors focus on the case when \(T\) is a piecewise monotone interval map with possible parabolic points and try to extend multifractal results obtained in [F. Hofbauer, Fundam. Math. 208, No. 2, 95–121 (2010; Zbl 1192.37056)] for the Lyapunov spectrum and the entropy Birkhoff spectrum by constructing the dimension Birkhoff spectrum. The authors are able to describe the intersections of the level sets with the set of points with positive lower Lyapunov exponent.