I think there is some misunderstanding here. Just like there is no such thing as a density of Cartesian points in direct space, there is no density of k-points in the reciprocal space. You can define k-points anywhere, like you can define a regular point anywhere in direct space. If you were to plot your band structure, your choice of k-points would not depend on your band model, but rather on the symmetries of your crystal (we usually plot along the high symmetry directions). How dense your sampling is, is an arbitrary choice (although too few points might of course not be enough to render the curves)
Rather, when we talk about a density in reciprocal space, we usually talk about density of states. That is to say how many states I get in the vicinity of a certain energy (the DOS is most often expressed as a function of energy), or, in the case of your question, in the vicinity of a certain k-point. It may seem like semantics, but I’m just making sure we’re on the same page. 359
I will assume that is what you meant.
If you’re looking for an expression of the DOS with respect to k, it actually does not depend on the assumptions you make for your band model. Let’s work in 1-D with periodic conditions on a crystal of length L (to simulate an infinite crystal). Recalling bloch theorem, our bloch wave at point k has the form: $$\psi_k(r)=u_k(r)e^{ikr}$$ With $u(r)$ having the peridocity of the lattice. The boundary conditions read: $$\psi_k(0)=u_k(0)=\psi_k(L)=u_k(L)e^{ikL}$$ By peridocitiy, $ u_k(0)=u_k(L)$, so that we must have $k=\frac{2\pi n}{L}$, with $n$ integer.
So the 'distance' in k-space separating two states is constant and equal to $\frac{2\pi }{L}$. The density in 1D is simply the inverse of that, so it reads: $$g(k)= \frac{L}{2\pi}$$ In fact, the DOS $dN/dk$ is homogeneous in k-space. In 3D, it generalizes as: $$g(\textbf{k})=\frac{V}{(2\pi)^3}$$ Your parabolic band only becomes relevant if you wanna translate that in a DOS in terms of energy (which we usually do, cause it would be a shame to keep things nice and simple and homogeneous, wouldn’t it?). The following is a derivation of the density of states per unit energy, valid for the 1D case only:
$$g(E)=\frac{dN}{dE}=\frac{dN}{dk} \frac{dk}{dE}=g(k) \frac{dk}{dE}$$ From your parabolic assumption on the conduction band (that I will downgrade to 1D for the purpose of this discussion): $$2k\frac{dk}{dE}\frac{h^2}{2m}=1$$ $$g(E)=g(k)\frac{m}{h^2k}=\frac{\sqrt{m}L}{2\pi h \sqrt{E-E_c}}$$ Be aware, that he result is very different for the 3D case (the derivation is a little different too since you have initially your density $ g(\textbf{k})$ expressed based on vectors). In 3D, you’ll find a DOS that is actually proportional to $\sqrt{E-E_c}$.
In short: The DOS “per unit of reciprocal distance in k space” $g(\textbf{k})$is homogeneous, and the shape of the band only matters when we wanna translate that per unit energy, $g(E)$ 976