Summary: The authors consider a class of Kukles planar polynomial differential systems of degree three having an invariant parabola. For this class of second-order differential systems, it is shown that for certain values of the parameters the invariant parabola coexist with a center. For other values they can coexist with one, two or three small amplitude limit cycles which are constructed by Hopf bifurcations. This result gives an answer to the question of X. Yang [Ann. Differ. Equations 7, No. 3, 323–363 (1991; Zbl 0747.34019)], about the existence of limit cycles for such a class of systems.