Nope, sorry to say. The easiest way to get arena tickets is to fight in the arena, not the pet derby. The arena can give up to 14 tickets, while the derby can give up to 3.
Well, technically if you hatch with a person who has a crow you have a very small chance of hatching a crow. The chance gets better if you use a pet that is just as rare or more rare than the crow. It is, still, a very slight chance. You are better off getting tickets.
