THEOREM. Assume that $L(t,q, \dot{q})$ is jointly $C^2$ in the considered coordinate patch and the Hessian matrix of coefficients $\frac{\partial^2 L}{\partial q^r \partial q^s}$ is everywhere non-singular.
Then, $p_k$ is a constant of motion (it is constant along every solution of EL equations) if and only if $q^k$ is cyclic ($\frac{\partial L}{\partial q^k}(t,q,\dot{q})=0$
for every choice of $t,q,\dot{q}$).
PROOF. If $L(t,q, \dot{q})$ is jointly $C^2$ and the Hessian matrix of coefficients $\frac{\partial^2 L}{\partial q^r \partial q^s}$ is non-singular, then for every choice of initial conditions $(t_0, q(t_0), \dot{q}(t_0))$ there is a local solution of EL equations satisfying those initial conditions (requiring for instance $C^3$ this solution turns out to be maximal and unique). Let us pass to the main statement.
If $\frac{\partial L}{\partial q^k}=0$ for every $(t, q, \dot{q})$, then every solution
$t \mapsto (t, q(t), \dot{q}(t))=:\gamma(t)$
of EL equations
$$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}}|_{\gamma(t)}\right)= \frac{\partial L}{\partial q^k}|_{\gamma(t)}\:, \quad \frac{dq^k}{dt}|_{\gamma(t)} = \dot{q}(t)$$
satisfies
$$\frac{dp_k|_\gamma(t)}{dt}=\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}}|_{\gamma(t)}\right)=0$$
so that $p_k$ is a constant of motion.
Vice versa, if for every solution $\gamma$ it holds
$$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{q}}|_{\gamma(t)}\right)=0\:,$$
form EL equations we also have that
$$\frac{\partial L}{\partial q^k}|_{\gamma(t)}=0\:.$$
To conclude, fix a kinetical state $(t,q,\dot{q})$. We know that there is a solution of EL equations which admits that state as initial conditions. Therefore, evaluating $p_k$ along that solution at time $t$,
$$\frac{\partial L}{\partial q^k}(t,q,\dot{q})\left(=\frac{dp_k|_\gamma(t)}{dt}\right)=0\:,$$
for every choice of $t,q,\dot{q}$. $\Box$
