We refer to one another as our dogs Mummy and Daddy because we find that amusing - however they are not child substitutes nor do we think of them as children exactly but they ARE family.
As far as a dog is concerned, the owners are sthe primary care givers and very much do fall into a parental role.
We make the decisions about whats safe, whats not safe, we provide the fun, the toys, the food, the equipment, the housing.... we determine where they can go and where they cannot.. clean up poo and pee and vom, are legally responsible for their actions/damage...
Very similar to parenting small children really!
That said, most small children are not shouting at wild rats under the shed in the rain at 1am!... (he's come in now. We don't use our shouty voices at 1am).
Frankly if more people saw themselves as a 'parent' rather than some sort of military style leader or ruler of the universe 'Master', we would have FAR fewer dog behaviour problems than we currently see.
That doesn't mean we should forget our dogs are dogs, tehy ARE dogs - but any good parent recognises their charges needs, desires, skill set and personality and provides for them and manages them as necessary.
So yes, I am my dogs parent, but no, they are not my children. Something I suspect you can't understand until you have dogs!