If the line does not intersect the bounding box of the parabola segment, you know that the line does not intersect the parabola. Otherwise, further checks may be required.
For example, there is a polynomial (obtained by multiplying the homogeneous coordinates of the line with the adjoint of the matrix of the parabola) which is zero if the line is a tangent to the parabola. If it is not zero, the sign tells us whether the line intersects the parabola or passes it. Or one could write the line as a parametric equation, plug that into the equation of the parabola and examine the discriminant of the resulting quadratic equation.
But all of this is assuming an infinite parabola, whereas a Bézier curve will usually be finite and probably isn't a real parabola either but just approximating one. Which means you'd likely want to put the parametric form of the Bézier curve into the equation of the line, compute the three solutions there and then see which of them are real and in the range $[0,1]$. Solving cubic equations is no fun, but it's probably the appropriate thing to do unless you are really dealing with exact parabolas.