This problem requires a strong understanding of parabola properties and the geometric transformation of reflection. We will first determine the directrix of the original parabola $P_1$ using its vertex and focus. Then, we will find the mirror image of this directrix with respect to the given line to obtain the directrix of parabola $P_2$.

1. Key Concepts and Formulas for Parabola Properties

The fundamental properties of a parabola relate its vertex, focus, and directrix:

• The vertex ($V$) of a parabola is the midpoint of its focus ($F$) and the point ($D'$) where the directrix intersects the axis of the parabola.
• The axis of the parabola is the line passing through the focus and the vertex.