In non parli a me di disperazione the verb is in the subjunctive mood. In the specific case, one cannot use the imperative, because the lei form is used: you are talking with someone not in familiar terms with you, so the verb must be in the third person and the imperative mood lacks it.
In the verb conjugations, the imperative is often showed with five voices (of course the first person singular doesn't make sense), but properly it has only two, like in Latin. The additional three forms are from the subjunctive.
It should be mentioned that the congiuntivo esortativo (exhortative subjunctive) is a common form, but here it's not the case. If the person you're talking to is in familiar terms, the sentence would be
Quindi, non parlare a me di disperazione
with the imperative.
As DaG remarks in a comment, the sentence could be interpreted as referring to a third person: Tizio is saying to Caio
Sempronio dice di essere disperato.
and Caio says to Tizio
La mia disperazione è immensa. Quindi, non parli a me di disperazione.
referring to Sempronio; in this case it would still be an imperative (with the substitute form from the subjunctive) but it could be interpreted as an exhortative subjunctive: the distinction is quite blurred.