Pre-treatment period in difference-in-differences model

I want to evaluate the consequences of a policy change using a diff-in-diff setup. I have quarterly data over ten years before the treatment ($t_{-10}$, $t_{-9}$, ..., $t_{-1}$) and ten years after the treatment ($t_{+1}$, $t_{+2}$, ..., $t_{+10}$) . However, in the pre-treatment period, it seems that the parallel trend assumption holds from $t_{-10}$ to $t_{-5}$ but not afterwards (i.e. from $t_{-5}$ to $t_{-1}$). It looks like the agents anticipate the policy change.

Is it possible in this case to only use the data from $t_{-10}$ to $t_{-5}$ for the pre-treatment period (and drop the observations from $t_{-5}$ to $t_{-1}$) in order to obtain an unbiased diff-in-diff estimate?

In a parallel trends test you will have to neglect one quarter of observations (which serves as the reference period) to which all others estimates are calculated in respect to. Usually, this is the period before the treatment starts. I assume that is $t_0$ in your case.
Given this, it depends on the setup of your policy evaluation. Did the agents have the relevant information to "see the treatment coming"? can you credibly rule out any other "treatment"? Maybe you can provide evidence that in $t_{-5}$ some news about the policy were released and you consider the "announcement date" rather than the "implementation date" as the appropriate event date?