The flow behavior of oil in nanochannels has attracted extensive attention for oil transport applications. In most, if not all, of the prior theoretical simulations, oil molecules were observed to flow steadily in nanochannels under pressure gradients. In this study, non-equilibrium molecular dynamics simulations are conducted to simulate the Poiseuille flow of oil with three different hydrocarbon chain lengths in graphene nanochannels. Contrary to the conventional perception of steady flows of oil in nanochannels, we find that oil molecules with the longest hydrocarbon chain (i.e., n-dodecane) exhibit notable stick-slip flow behavior. An alternation between the high average velocity of n-dodecane in the slip motion and the low average velocity in the stick motion is observed, with a drastic, abrupt velocity jolt of up to 40 times occurring at the transition in a stick-slip motion. Further statistical analyses show that the stick-slip flow behavior of n-dodecane molecules originates from the molecular alignment change of oil near the graphene wall. The molecular alignment of n-dodecane shows different statistical distributions under stick and slip motion states, leading to significant changes of friction forces and thus notable velocity fluctuations. This work provides new insights into the Poiseuille flow behavior of oil in graphene nanochannels and may offer useful guidelines for other mass transport applications.