Great work and great post. Your method is the way to do it. My answer is more of an elaboration about the geometric interpretation of this parabola. I hope it's useful.
$$|z-1| = Re(z+1)$$
We can read this as "the distance of $z$ from $1$ is equal to one more than the real part $x$ of $z$". Incidentally, this is directly associated with the geometric definition of a parabola.
A parabola is the set of all points that are equidistant from one point (the focus) and a line (the directrix). Let the parabola have vertex $h+ik$. Now define $c$ to be the distance from the vertex to the focus; then $c$ is also the perpendicular distance from the vertex to directrix. If the parabola opens to the right, the focus is $h+c+ik$. Then geometrically,
$$|z-(h+c+ik)| = |z - (h-c+iy)|$$
$$|z-h-c-ik)| = |z - h+c-iy)|$$
Now compare with your parabola. It's vertex in rectangular coordinates is $(0,0)$.
$$|z-c| = |z+c-iy|$$
It appears that $c=1$ and thus your focus is $(1,0)$. Likewise, the distance from the directrix is given by the real part of $z$. The directrix itself is the line $x=-1$. This information was encoded in the original equation all along!