the setence could equally well be 'commenca'
Yes, "elle commença à parler au chien", but apparently Duolingo has decided to ignore the French simple past tense.
Why is there an à before parler in this sentence but not, for example, in "Elle a pu parler?
For the same reason in both languages: pouvoir/can are modal verbs, constructed without a preposition: elle peut parler = she can talk
Commencer à + infinitive verb = start to + verb
Pouquoi pas "parler avec le chien" mais "parler au chien"
Probably because the dog can't answer, atleast mine can't, thus you talk to it not with it