Oh no, it’s possible that they all can get along, but they have to have a particular chemistry, just like humans do with each other. Sometimes, cat pokémon and dog pokémon adore one another, and sometimes, they go at it like, well, cats and dogs. And sometimes, the latter occurs for no good reason at all.
Of course, other times, there may be a particular reason, so it would be worth it to observe your pokémon and see if there is a behavior that causes them to interact this way. It’s likely that your lillipup simply needs to be trained to not chase your delcatty, while your purugly needs to be trained to use better problem resolution skills.