it's not sure that it is a parabola. It could also be an hypernola with non-orthogonal asymptotes.
So, I suggest a quadratic curve regression with least squares fitting.
After computing the coefficients for the general equation of the quadratic curve, it is easy to identify a parabola or an hyperbola or etc. Then, the center, the axis,... can be determined and asympotes as well if they are some.