Parabolic Components in Cubic Polynomial Slice $Pe{r}_1(1)$

We prove that the boundary of every parabolic component in the cubic polynomial slice $Pe{r}_1(1)$ is a Jordan curve by adapting the technique of para-puzzles presented in \cite{Roesch1}. We also give a global description of the connected locus ${\mathcal{C}}_1$: it is the union of two main parabolic components and the limbs attached on their boundaries.