Parallel Trend assumption in DiD This assumption should rule out the concern of selection bias, right? For example, I am looking at the effect of increasing police presence on crimes. The government decided to put more police in certain areas. Of course, the police will be placed into cities with more violence. If 2 cities have similar trend in crime pre-treatment, I don't have to worry about the selection bias, right?
 A: As you can see here, the parallel trend assumption does not require that the pre-treatment response trends are "similar" between the two groups. They need to be parallel in time, whatever your expression of trend is (usually linear). Of course, if the time-trends are equal (note: "similar" is too imprecise) they are, of course parallel. See figure 1 from link below.
Selection bias is a whole other can of worms. The point of selection bias is that it affects observable trends. As a consequence the data do not tell you if there's bias, rather the design does. It is in fact possible for selection bias to cause non-parallel time-trends to be parallel. Consider comparing police forces in municipalities between two regions; say you are curious if there are protocols implemented for investigating homicide. If you only sample units of a certain size, you guarantee the level of crime has some degree of preparedness for particular issues, notably murder. But if one states comprises mostly smaller municipalities where forces do not have protocols since the incidence is much lower, your design has an incorrect inference that the two groups are "similar" (equal, parallel, or otherwise).

