But parents should be penalized/inconvenienced if they can't control their children, not the store/website.

In some states, bounty hunters can find violators of various laws and bring them to the state in exchange for money. Allowing bountry hunters to be on the lookout for underagers trying to enter stores and report them could be a profitable endeavor for both the bounty hunters and the state, providing a market-based incentive to protecting children.

Stores/websites should only be penalized if they are specifically targeting and inviting children to enter.

Certainly there are some types of stores that are very safe for children to enter. So, the exception should be structured like this: "Stores that only sell these items mean that parents will not be fined if children enter it unaccompanied." Additional conditions could be attached, e.g. "a safe store shall verify the ID of anyone purchasing alcohol." And maybe other benefits can be attached to being a "safe" store, like tax incentives, etc. If the store violates that condition, then it should pay a fine or lose any other benefits of being considered a "safe" store. But a business should have a right not to be a "safe store" and the duty to prevent children from entering those should 100% fall on the parents.