I suspect we're not getting the full details here or talking about the same situation especially as you're talking about your parked car being hit and the OP is talking about getting their insurance company to pay and or help out on a coverage they don't have on their policy. I'm talking about responsibilities of a car insurance company as it relates to your comment about "you pay the premium that's what they are there for", just paying premium does not entitle you to this.
Your premium is comprised of coverages you pay for (legally required and optional), the dollar amount is comprised of many other components. Just paying your policy premium does not entitle you to have your insurance company handle a claim on your behalf for a coverage you don't have. We aren't talking about medical and whatnot that falls under liability.
People can confuse things that insurance companies do because they are trying to have good PR and things that companies are required to do. I have seen enough claims in my time where the insurance company did things they were not required to do, but that is not to be confused with merely calling up your insurance company and telling them "I pay you premiums so handle this for me". You have to be understanding to what coverages you are actually paying for.