Elderly people often go into a home because they can no longer look after themselves, this is often a very difficult decision on their children's end, but looking after an elderly person is a VERY difficult thing to do in itself, which can put a huge strain on their relationship especially when the elder needs 24 hour care which is impossible for their child to do if they have a full time job or even a family of their own. Many elderly people would agree that they need professional care rather than robbing their own child's freedom to take care of them on a full time basis... As someone who has witnessed my own mother's and grandmother's relationship degrade in just five years of my grandmother living with us, I can honestly say that the family is very strained because of how difficult it is living with her, let alone caring for her.
Pets are usually able to take care of themselves for a good portion of the day (ie. When their owner is at work), their level of care is a lot lower than a dependable elder, and pets can adapt to family life much easier, not only that but they are proven to be great stress relievers and pets such as dogs are great to have around for peace of mind to protect the family/house.
The fact that you're likening a difficult situation like taking care of an elderly person full time to owning a pet shows that you are quite ignorant.