is there an algorithm to rotate a parabola about the origin?

Yes. It follows from the formula to rotate a single point about the origin. Given point P(x,y), if you rotate that by an angle θ (positive meaning counter-clockwise) about the origin, it will move to a new location. Call that point P'(x',y'). The formula for finding x' and y', given x, y and θ is:

x' = x cos θ - y sin θ

y' = x sin θ + y cos θ

From that, you can work out that the inverse relation is:

x = x' cos θ + y' sin θ

y = -x' sin θ + y' cos θ

Suppose your equation for the parabola before rotation is y = x². (Keep it simple.) In terms of x' an y', that relationship is:

-x' sin θ + y' cos θ = (x' cos θ + y' sin θ)²

Dropping the ' primes on x and y, that simplifies to

x² cos² θ + 2xy (cos θ)(sin θ) + y² sin² θ + x sin θ - y cos θ = 0

I moved everything to one side because of the xy term. It's not easy to untangle that into a formula for either x or y. (You can do it, using the quadratic formula, but the result isn't very pretty

As a specific example, if θ is 45 degrees (pi4 radians) then sin θ = cos θ = 1/√2. The equation for that parabola would be:

x²/2 + 2xy/2 + y²/2 + x/√2 - y/√2 = 0

You can see a graph of that at Wolfram Alpha:

http://www.wolframalpha.com/input/?i=x%C2%B2%2F2+%...

This makes no sense.