So, I have this:
Is it correct? And what does that に particle mean?
EDIT: I have had a misunderstanding as reported by chocolate in the comments, so I have corrected myself.
The に particle here shows the agent (the one who does the action). 犬を追いかける is "to run after a dog".
So 「父に犬を追いかけられています。」 is "The dog is run after by my father."
I think 父が犬に追いかけられています is correct.
In the case of 父に犬を追いかけられています, if I must say, it means 'My dog gets chased by my father.'
に is a postpositional particle for an object.
I have my father call a dog.