In this work, we have modified a scenario, originally proposed by Grimus and
Lavoura, in order to obtain maximal values for atmospheric mixing angle and $CP$ violating Dirac phase of the lepton sector. To achieve this, we have employed $CP$ and some discrete symmetries in a type II seesaw model. In order to make predictions about neutrino mass ordering and the smallness of the reactor angle, we have obtained some conditions on the elements of the neutrino mass matrix of our model. Finally, within the framework of our model, we have studied quark masses and mixing pattern.