Actually, a model such as $Y = b_0 + b_1X + b_2X^2$ is not a 3D parabola, but a 2D parabola. There are only two variables ($Y$ and $X$), in other words, the function is still $Y = f(X)$.
A 3D parabola would be a paraboloid, thus the model would be $Y = b_0 + b_1X + b_2X^2 + b_3Z + b_4Z^2$, and a function of the type $Y = f(X,Z)$.
Alternatively, there are mixed models such as $Y = b_0 + b_1X + b_2X^2 + b_3Z$ , which would be a Parabolic Cylinder in 3D and also $Y = f(X,Z)$.
As @Evator mentioned, all these models are in fact linear models, where the term linear is referred to the coefficients, not the variables. Thus, linearity in variables is different from linearity in coefficients. These models fit quite well but sometimes there is a cost: multicollinearity and increased variance.