Parallel trends basically says that the treated units, in the absence of treatment, would have evolved like the untreated units. If that is the case, DiD identifies the treatment effect.
More specifically, we consider a model like
$$
y=\beta_0+\delta_0d_2+\beta_1d_T+\delta_1d_2\cdot d_T+\textit{other factors}+u,
$$
where $d_T=1$ for the treated, and $d_2=1$ for the post-policy period.
Schematically:
\begin{pmatrix}
&Before&After&After - Before\\\hline
Control&\beta_0&\beta_0+\delta_0&\delta_0\\
Treatment&\beta_0+\beta_1&\beta_0+\delta_0+\beta_1+\delta_1&\delta_0+\delta_1\\\hline
Treatment - Control &\beta_1& \beta_1+\delta_1&\delta_1
\end{pmatrix}
Now, if $\delta_0$ differed between treated and untreated, the above two-way fixed effects model would no longer identify $\delta_1$.
For example, Wooldridge's textbook (if I remember correctly) cites a study on rumors that a garbage incinerator would be built in Andover, Mass. beginning after 1978. Building started in 1981. We'd
conjecture that house prices close to the incinerator fall. Define $\textit{nearinc}=1$ if a house is closer to the site
than 3 miles. For 1981 data, estimating the model
$$
\textit{realprice}=\gamma_0+\gamma_1\textit{nearinc}+u
$$
gives
$$
\widehat{\textit{realprice}}=\underset{(2,951.19)}{101,307.5}-\underset{(6,219.26)}{30,688.27}\textit{nearinc}\qquad n=142
$$
Does that mean the incinerator caused house prices to decrease by 30,000\$? No, because houses close to the site
may be of lower quality (such sites aren't built where the mayor himself lives, no ;-)?), a factor contained in $u$.
To see this, we can redo the regression for 1978, before the start of the rumors:
$$
\widehat{\textit{realprice}}=\underset{(1,881.16)}{82,517.23}-\underset{(5,992.56)}{18,824.37}\textit{nearinc}\qquad n=179
$$
That is, houses close to the site were indeed already of lower value prior to the rumors. A more useful causal effect
is obtained by comparing the effect of nearness before and after the rumors:
$$
\hat{\delta}_1=-30,688.27-(-18,824.37)=-11,863.9
$$
$\hat{\delta}_1$ a DiD estimator, because it can be written as
\begin{align*}
\hat{\delta}_1&=(\overline{\textit{realprice}}_{near,1981}-\overline{\textit{realprice}}_{far,1981})-(\overline{\textit{realprice}}_{near,1978}-\overline{\textit{realprice}}_{far,1978})\\
&=(\overline{\textit{realprice}}_{near,1981}-\overline{\textit{realprice}}_{near,1978})-(\overline{\textit{realprice}}_{far,1981}-\overline{\textit{realprice}}_{far,1978})
\end{align*}
Now, if price trends in close and non-close neighborhoods were diverging irrespective of the rumors regarding the incinerator (think of, say, a high-paying employer moving into town at the same time, such that many new well-paid employees drive up prices in "good" neighborhoods far away from where a mayor would build an incinerator), we could not use the DiD estimator to identify the causal impact of such an incinerator on house prices (which could for example be used to compensate people living close to the site for the loss in value of their property).
