This is not an easy question to answer since the system is quite complex. Besides, the way the stick will bend depends on the type of stress you submit it to, in other words, what the boundary conditions of the system are.
A simple model for a beam of length $\ell$ is given by the following differential equation
$$\frac{d^4 Y}{dx^4} = \frac{W(x)}{EI} \; \text{ for } 0 < x < \ell \; ,$$
in which $Y$ is the vertical displacement of the beam if it is aligned horizontally. $W(x)$ is the vertical load per unit length, that is essentially the force to which the beam is submitted. Note that since this model is 1D, it doesn't account for torsion or shears along other directions than the vertical one. $E$ is Young's modulus of elasticity and $I$ is the moment of inertia of a cross section of the beam about the axis. Depending on boundary conditions, one can describe different types of problems:
- $Y=Y'=0$ for a clamped beam.
- $Y=Y''=0$ for a hinged or simply-supported ends.
- $Y''=Y'''=0$ for free ends.
This equation is known as the Euler-Bernoulli equation for beams.
So, depending on what you want to model exactly, you'll have to solve this equation for an appropriate choice of boundary condition and load $W(x)$. For instance, you could take one clamped end and another free end on which a load is concentrated to simulate a hand holding a beam still while the other is trying to push an end. In this case, the solution will not be given by a parabola but by a third degree polynomial.
This does nothing yet to solve the problem for more complicated stresses, for instance along other directions than the vertical or for torsions. From there on, this really becomes an engineering question. If you want to delve more deeply into the subject, you should read up on materials science. One thing is clear though, the more complex the system, the less likely you'll obtain a conic section while bending.