The Berry curvature is defined as:
$$
\Omega_{\mu\nu}(\mathbf{k})=\partial_{\mu}A_{\nu}(\mathbf{k})-\partial_{\nu}A_{\mu}(\mathbf{k}),
\tag{1}
$$
where $A_{\mu}(\mathbf{k})=\langle u_{\mathbf{k}}|i\partial_{\mu}u_{\mathbf{k}}\rangle$ is the Berry connection, $|u_{\mathbf{k}}\rangle$ is a Bloch state, and $\partial_\mu\equiv \frac{\partial}{\partial k_\mu}$, and $\mu,\nu=x,y,z$.
Invesion symmetry. Under inversion, $\mathbf{k}\to-\mathbf{k}$, so that applying the inversion operation $\mathcal{I}$ on a Bloch state gives $
\mathcal{I}|u_{\mathbf{k}}\rangle=|u_{-\mathbf{k}}\rangle$. If the system is invariant under inversion, then $|u_{\mathbf{k}}\rangle$ and $|u_{-\mathbf{k}}\rangle$ must be the same state up to a global phase, so that:
$$
\mathcal{I}|u_{\mathbf{k}}\rangle=e^{i\varphi_{\mathbf{k}}}|u_{\mathbf{k}}\rangle\Longrightarrow |u_{-\mathbf{k}}\rangle=e^{i\varphi_{\mathbf{k}}}|u_{\mathbf{k}}\rangle.\tag{2}
$$
For the Berry connection, $\mathcal{I}A_{\mu}(\mathbf{k})=A_{\mu}(-\mathbf{k})$. If the system has inversion symmetry, then
$$
\begin{eqnarray}
A_{\mu}(-\mathbf{k})&=&\langle u_{-\mathbf{k}}|i\partial_{\mu}u_{-\mathbf{k}}\rangle \tag{3}\\
&=& \langle u_{\mathbf{k}}|e^{-i\varphi_{\mathbf{k}}}i\partial_{\mu}\left(e^{i\varphi_{\mathbf{k}}}u_{\mathbf{k}}\right)\rangle \tag{4}\\
&=& \langle u_{\mathbf{k}}|e^{-i\varphi_{\mathbf{k}}}ie^{i\varphi_{\mathbf{k}}}\partial_{\mu}u_{\mathbf{k}}\rangle + \langle u_{\mathbf{k}}|e^{-i\varphi_{\mathbf{k}}}i^2e^{i\varphi_{\mathbf{k}}}u_{\mathbf{k}}\rangle\partial_{\mu}\varphi_{\mathbf{k}}\tag{4} \\
&=& \langle u_{\mathbf{k}}|i\partial_{\mu}u_{\mathbf{k}}\rangle -\partial_{\mu}\varphi_{\mathbf{k}} \tag{5}\\
&=&A_{\mu}(\mathbf{k})-\partial_{\mu}\varphi_{\mathbf{k}},\tag{6}
\end{eqnarray}
$$
where in the second line I used the result for the Bloch state in a system with inversion symmetry, and in the third line the chain rule for differentiation. This result means that for a system that is invariant under inversion, then $A_{\mu}(\mathbf{k})$ and $A_{\mu}(-\mathbf{k})$ differ at most by a gauge transformation.
We are now ready to look at the Berry curvature. Under inversion, $\mathcal{I}\Omega_{\mu\nu}(\mathbf{k})=\Omega_{\mu\nu}(-\mathbf{k}$). If the system has inversion symmetry, then
$$
\begin{eqnarray}
\Omega_{\mu\nu}(-\mathbf{k})&=&\partial_{\mu}A_{\nu}(-\mathbf{k})-\partial_{\nu}A_{\mu}(-\mathbf{k}) \tag{7}\\
&=&\partial_{\mu}\left(A_{\nu}(\mathbf{k})-\partial_{\nu}\varphi_{\mathbf{k}}\right)-\partial_{\nu}\left(A_{\mu}(\mathbf{k})-\partial_{\mu}\varphi_{\mathbf{k}}\right) \tag{8}\\
&=&\partial_{\mu}A_{\nu}(\mathbf{k})-\partial_{\nu}A_{\mu}(\mathbf{k})-\partial_{\mu}\partial_{\nu}\varphi_{\mathbf{k}}+\partial_{\nu}\partial_{\mu}\varphi_{\mathbf{k}} \tag{9}\\
&=&\partial_{\mu}A_{\nu}(\mathbf{k})-\partial_{\nu}A_{\mu}(\mathbf{k}) \tag{10}\\
&=&\Omega_{\mu\nu}(\mathbf{k})\tag{11},
\end{eqnarray}
$$
where in the second line I used the result for the Berry connection in a system with inversion symmetry. This proves that for a system with inversion symmetry, $\Omega_{\mu\nu}(\mathbf{k})=\Omega_{\mu\nu}(-\mathbf{k})$.
Time reversal symmetry. You can use an analogous procedure (I encourage you to try) to prove that for a time reversal invariant system, $\Omega_{\mu\nu}(\mathbf{k})=-\Omega_{\mu\nu}(-\mathbf{k})$. All you need to know is how the time reversal operator acts on a Bloch state, $\mathcal{T}|u_{\mathbf{k}}\rangle=|u_{\mathbf{-k}}^{\ast}\rangle$, and the rest of the proof proceeds in the same way.
Physical interpretation. Berry phase-like quantities look at the evolution of Bloch states at neighbouring $\mathbf{k}$-points in the Brillouin zone. As an example, the Berry connection is looking at the overlap between a state $|u_{\mathbf{k}}\rangle$ and a state infinitessimally away from it, $\partial_{\mu}|u_{\mathbf{k}}\rangle$. As such, they are useful for calculation properties that depend on the structure of the Block states across the Brillouin zone. A well-known example is the calculation of topological invariants of materials, which measure the "twists" that the electronic wave function has when crossing the Brillouin zone. I am not familiar with applications in valleytronics, so will leave that for someone more knowledgeable.
Futher reading. An excellent book to learn about Berry phase-like quantities and applications (modern theory of polarization, topological materials, etc.) is David Vanderbilt's book.
