Many physical laws are formulated as differential equations. Parabolic equations have parabola as a solution, elliptic equation an ellipse, hyperbolic an hyperbola.
All these solutions are curves, not straight lines. Is there any physical intuition behind this?
A straight line is a solution of $y^{''}=0$. The shortest answer to your question is "not all laws look like that". (In any case, whether an equation looks like that depends on the coordinate system used.)
Let's review what the equation types you mentioned actually mean, because they're not named for their solution shape. (That wouldn't work in general, because the shape is often either nameless, unknown or solution-dependent.)
Linear equations are those for which, if $f,\,g$ are solutions, so is $pf+qg$ for constants $p,\,q$ provided $p+q=1$. We call the equation homogeneous if, more generally, any constants $p,\,q$ work. The name "linear" derives from the fact that the closure rule for a homogeneous linear equation's solution set is similar to that of a straight line through the origin, while the closure rule for a general linear equation's solution set is similar to that of an arbitrary straight line.
Ellipses, parabolas and hyperbolas are collectively known as conic sections, and they each have an equation of the form $\sum_{ij}A_{ij}y_iy_j + \sum_{i}B_iy_i + C = 0$ with $A$ a constant matrix, $\mathbf{B}$ a constant vector and $c$ a constant scalar. The ellipse-parabola-hyperbola distinction concerns whether $\det A$ is positive, zero or negative. We similarly subdivide differential equations of the form $\sum_{ij}A_{ij}\partial_i\partial_j y + \sum_{i}B_i\partial_i y + Cy + D = 0$, with all coefficients $y$-independent (though in general they can be functions of other variables $x^i$, viz. the notation $\partial_i :=\frac{\partial}{\partial x^i}$). Hence such equations can be elliptic, parabolic or hyperbolic.
Hyperbolic differential equations are a slightly more general class of differential equation than the above discussion implies, as the inclusion of lower-order terms makes little difference to the solutions' qualitative behaviour. In particular, hyperbolic equations' solutions have "well-posed initial value problems", meaning exactly one solution exists once you specify suitable constraints somewhere on the solution and its first-order derivatives. This is very important in non-quantised physical models of spacetime, because we expect the past to determine the present, and the present to determine the future.
However, this kind of misunderstanding of terminology is understandable. A former colleague of mine was once under the misapprehension, as a professor, that galileons were galilean-invariant. As with the examples above, equations' solutions are called "galileons" due to the equation's form being reminiscent of its namesake, in this case Galilean transformations.
