In this talk, I will present a work in progress with Matthieu Astorg and Lorena Lopez-Hernanz. We are interested in studying holomorphic endomorphisms of $\mathbb{C}^2$ which are tangent to the identity at the origin, and our goal is to understand how the dynamics changes when we perturb such maps. In particular, we generalize the results obtained by Bianchi and show a result "à la Lavaurs" when the unperturbed map admits a basin parabolic centered in a characteristic direction, but it does not fix a complex line.