The paper is devoted to study the one parameter family of cubic polynomials \[ g_b(z)= \lambda z+ bz^2+z^3, \quad b\in \mathbb{C},\tag{1} \] where \(\lambda= e^{2\pi i\theta}\) is a fixed complex number of modulus 1. The authors show that the bifurcation locus of (1) contains quasi-conformal copies of the quadratic Julia set \(J(\lambda z+z^2)\). As a consequence, they show that when the Julia set \(J(\lambda z+z^2)\) is not locally connected, then the bifurcation locus is not locally connected. The proofs use results of Douady-Hubbard [see \textit{A. Douady} and \textit{J. H. Hubbard}, C. R. Acad. Sci., Paris, Sér. I 294, 123-126 (1982; Zbl 0483.30014)] and Branner-Hubbard [\textit{B. Branner} and \textit{J. H. Hubbard}, Acta Math. 160, 143-206 (1988; Zbl 0668.30008)].