Power increases the amount of times you are able to cheer for your pet and how fast your pets cheer energy refills in pet vs pet. It also increases the amount of damage your pet's powers do in pet vs pet and regular fights. Hope this helps!
Someone once told me in the morph tent that pedigree didn't matter. He was looking for high level talents and powers, yet thought pedigree was unimportant. It means the higher the pedigree, the more high level talents and powers a pet can potentially get. Power is great to have! I can't say for sure, but it might also help with epic hits by pets.