A parabola has a quadratic (highest power is x^2 or y^2, but not both, and there can be multiples of these squares) in one variable only (usually x) with the other variable (usually y) equal to the quadratic. An upright or inverted parabola (U shaped or inverted U shaped) consists of y= quadratic in x; a parabola on its side, pointing right or left, consists of x= quadratic in y. The parabola will have a symmetrical shape where the axis of symmetry acts like a mirror reflecting the two halves or arms of the parabola in one another.
Examples: y=4x^2, 2y=9-x^2, y=6x^2+5x-4, x=3y^2+1, 2x+3y^2=10, y+3x^2=4x-5, etc.