Since these are parabolas, $a$ and $c$ must be nonzero. Let $x = a^{2/3} c^{1/3} X$ and $y = a^{1/3} c^{2/3} Y$. Under this scaling, the equations become $X^2 = Y + B$ and $Y^2 = X + D$ where $B = b a^{-4/3} c^{-2/3}$ and $D = d a^{-2/3} c^{-4/3}$. Now substituting $Y = B - X^2$ into $Y^2 = X + D$ we get the fourth-degree equation $X^4 - 2 B X^2 - X + B^2 - D = 0$. The discriminant of this, according to Maple, is $-256\,{B}^{3}+288\,B D -27+256\,{B}^{2}{D}^{2}-256\,{D} ^{3}$. The curve where the discriminant is $0$ separates the $BD$ plane into three regions like this:
We have: no real solution in the red region, one on the red-yellow boundary, two in the yellow region, three on the yellow-blue boundary (except at the sharp cusp $B=D=3/4$ where there are two), and four in the blue region.
