"Parents buy an expensive dog," this part is correct but usually when speaking about parents you want to specify whose parents. e.g. "My parents..." or "Their parents..." or "Her parents..." You don't Have to include this but it sounds funny when you don't.
"regard it as their another daughter," Their and another cannot go together.
1. You could say, "regard it as their daughter" and just leave out another. This would be less specific because it doesn't say whether or not they have any other children.
2. You could say, "regard it as another daughter" and leave out their. Since you are already talking about parents you don't have to include "their" it can just be assumed by context.
3. You could say, "regard it as their other daughter." This would be the most specific because it implies that they already have one daughter and now are adding the dog with her.
"and stay with it wherever they are." If you are talking about the dog staying with them you should say, "and it stays with them wherever they go." If you are focusing on the parents always keeping the dog with them you could say, "and they keep/bring it with them wherever they go." The reason you should use "go" at the end of the sentence and not "are" is because "staying with them wherever" implies the parents moving from place to place.
And yes, you can use three verbs in one sentence. Hope this helps