Thank you for the link, Jaap. That article was very helpful despite the fact that I don't speak French. But now I have another (slightly more relevant) question: What are the equation(s) of the parabola(s) that pass through an arbitrary point within the "parabola of safety?" I think I can figure out how to determine the initial angle from that equation.
Check out the graph that shows the line OC passing through the focus of the parabola. It should be easy to compute the parabola by first computing the location of the focus (the point the horizontally fired object (blue curve) crosses the OC line) and then from the focus and points O and C computing the resulting parabola.