In many fields such as computer animation, computer graphics, machine vision, robotics, virtual reality and CAD/CAM, it is a common problem to detect the positional relationship between two planar conics. Based on the generalized characteristic polynomial and the affine invariants of two conics, explicit conditions are derived for classifying the various positional relationships between an ellipse and a parabola, namely, separation, exterior tangency, intersection, interior tangency and inclusion. The explicit conditions can be expressed in terms of the coefficients of the quadratic forms representing the ellipse and the parabola.