By a result of G. Pareschi [Math. Ann. 291, No. 1, 17–38 (1991; Zbl 0723.14004)] smooth curves from the same linear system on a smooth del Pezzo surface (i.e. smooth surfaces with ample anticanonical class) of degree \(\geq 2\) have the same gonality, with one exception due originally to R. Donagi and D. R. Morrison [J. Differ. Geom. 29, No. 1, 49–64 (1989; Zbl 0626.14006)]. Later A. L. Knutsen [Math. Nachr. 256, 58–81 (2003; Zbl 1048.14015)] has studied the same question for smooth del Pezzo surfaces of degree 1, by classifying all exceptions. In this paper the authors prove that the result of Pareschi still takes place also for smooth weak del Pezzo surfaces (i.e. smooth surfaces with nef and big anticanonical class) of degree \(\geq 2\) (for a classification of weak del Pezzo surfaces see e.g. [E. Ballico, G. Casnati and R. Notari, “Canonical curves with low apolarity”, preprint, arXiv:1003.3035]. The proof follows and extends the argument of Pareschi from del Pezzo surfaces to weak del Pezzo surfaces. It will be interesting to extend also the result of Knutsen for weak del Pezzo surfaces of degree 1.