This paper is devoted to an initial-boundary value problem for a class of $p$-Kirchhoff type parabolic equations. Firstly, we consider this problem with a general nonlocal coefficient $M(\Vert \mathrm{\nabla }u{\Vert }_p^p)$ and a general nonlinearity $k(t)f(u)$. A new finite time blow-up criterion is established, also, the upper and lower bounds for the blow-up time are derived. Secondly, we deal with the case that $M(\Vert \mathrm{\nabla }u{\Vert }_p^p)=a+b\Vert \mathrm{\nabla }u{\Vert }_p^p$, $k(t)\equiv 1$ and $f(u)=|u{|}^{q-1}u$, which was considered by Li and Han [Math. Model. Anal. 2019; 24: 195-217] only for $q>2p-1$. The threshold results for the existence of global and finite time blow-up solutions to this problem are obtained for the case $1<q\le 2p-1$, which, together with the results given by Li and Han, shows that $q=2p-1$ is critical for the existence of finite time blow-up solutions to this problem. These results partially generalize and extend some recent ones in previous literature.