What happens if the parallel trend assumption between control and treatment groups (diff-in-diff) or between individuals (individual-level fixed effects model) is not fulfilled?
Andrew Gelman, a prominent Professor of Statistics and Political Science has a recent post that you may find useful.
But, first, I recommend you examine his earlier post from a few years back about how Difference-in-difference estimators are a special case of lagged regression.
He makes the argument that:
[Difference in Difference] just a special case of lagged regression where the lag is restricted to have a coefficient of 1. In educational research, this is sometimes called the analysis of “gain scores.”
NOTE: As an aside, he suggests that lagged-regressions may "just be better than difference-in-difference," since DiD may "limit your statistical efficiency and range of applicability,":
That said, as Gelman acknowledges, difference-in-difference models can be useful when you have "a model with error terms for individual units" since "differencing makes the error terms drop out" and thus can give a "cleaner estimator."_
This aside should be read in the context of the question you asked - namely, how to respond when the assumptions underlying Diff-in-Diff are not satisfied.
For this, please read the aside above n the context of the follow-up post update this May by Gelman which quotes a paper addressing your concern:
Angrist and Pischke (2009) show that difference-in-differences and the lagged-dependent-variable regression estimates have a bracketing relationship. Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming the parallel trend will overestimate the effect; in contrast, if the parallel trend is correct, then mistakenly assuming ignorability will underestimate the effect.
So, as they say: "mistakenly assuming the parallel trend will overestimate the effect" - and, in such cases, the lagged-dependent variable version of the DiD regression model may be preferable. It is recommended you test your data for these assumptions and select between these two options accordingly.