Summary: Given an \(n\)-vertex graph \(G\) with a fixed linear ordering of \(V(G)\) and two integers \(k\), \(b\), the problem fixed-order book drawing with few crossings per edge asks whether or not \(G\) admits a \(k\)-page book drawing where the maximum number of crossings per edge can be upper bounded by the number \(b\). This problem was posed as an open question by S. Bhore et al. [J. Graph Algorithms Appl. 24, No. 4, 603–620 (2020; Zbl 1451.05222)]. In this paper, we study this problem from the viewpoint of parameterized complexity, in particular, we develop fixed-parameter tractable algorithms for it. More specifically, we show that this problem parameterized by both the bound number \(b\) of crossings per edge and the vertex cover number \(\tau\) of the input graph admits an algorithm with running time in \((b+2)^{\mathcal{O}(\tau^3)}\cdot n\), and this problem parameterized by both the bound number \(b\) of crossings per edge and the pathwidth \(\kappa\) of the vertex ordering admits an algorithm with running time in \((b+2)^{\mathcal{O}(\kappa^2)}\cdot n\). Our results provide a specific answer to Bhore et al.’s question [loc. cit.].