The words "elliptic", "parabolic" (or "like a parable"), "hyperbolic", and "circular" all have meaning in rhetoric. Are these meanings etymologically connected to the conic sections?
It appears that the mathematical meanings follow the non-mathematical ones, in the sense that the conic section names are based on metaphorical usage of pre-existing terms from the Greek. For example, in the ellipse entry mentioned above, we see:
Ellipse, 1753, from Fr. ellipse (17c.), from L. ellipsis "ellipse," also, "a falling short, deficit," from Gk. elleipsis (see ellipsis). So called because the conic section of the cutting plane makes a smaller angle with the base than does the side of the cone, hence, a "falling short." First applied by Apollonius of Perga (3c. B.C.E.).
Ellipsis, 1560s, "an ellipse," from L. ellipsis, from Gk. elleipsis "a falling short, defect, ellipse," from elleipein "to fall short, leave out," from en- "in" + leipein "to leave" (see relinquish). Grammatical sense first recorded 1610s.