This post simplifies how to conduct a pre-trend analysis using "coefficient plotting", which follows from lecture notes by Pischke 2005.
I simplified copying the description here:
A formal test which is also suitable for multivalued treatments or several groups is to interact the treatment variable with time dummies. Suppose you have 3 pre-treatment periods and 3 post-treatment periods, you would then regress
$$y_{it} = \lambda_i + \delta_t + \beta_{-2}D_{it} + \beta_{-1}D_{it} + \beta_1 D_{it} + \beta_2 D_{it} + \beta_3 D_{it} + \epsilon_{it}$$
where $y$ is the outcome for individual $i$ at time $t$, $\lambda$ and $\delta$ are individual and time fixed effects (this is a generalized way of writing down the diff-in-diff model which also allows for multiple treatments or treatments at different times). If the outcome trends between treatment and control group are the same, then $\beta_{-2}$ and $\beta_{-1}$ should be insignificant, i.e. the difference in differences is not significantly different between the two groups in the pre-treatment period.
So, it seems that one way to satisfy pre-trend testing is when the estimates of $\beta_{-2}$ and $\beta_{-1}$ are insignificant. But what about $\beta_j$, $j \geq 0$? From Pischke's notes, it seems that they should be identical but he did not mention whether they should be significant or not.
Apart from that, in situations where $\beta_{-2}$ and $\beta_{-1}$ are significant but close to zero, can it be counted as a parallel trend as Pischke (2005) as above?