I'd do this with two content types, rather than webform:
- Available slot-dates
- Sign-ups
You create the content for the available slot-dates (one node per date-slot), parents create the sign-up content. The Sign-up content includes the basic information that you need for kid's background context. After, you can add a paragraph to this content type that allows parents to specify the week that they sign their kids up for. Do this through an entity reference to your content, Available Slot-Dates. The benefit of this is that paragraphs on the node/add form allows the user to add another paragraph (in your case, this lets them select another week and/or add additional kids, without seeing extra form fields, or being limited to 5 kids/submission).
Then, you can track which of the Available Slot-Dates have been claimed by creating a rule that changes a field on the available slot-dates content that was referenced by the parent-created sign-ups.
Finally, you can only show parents dates that are available by creating a view for the entity reference. Filter for available slot-dates that are still available using the indicator field from the prior step. If you use this view on the entity reference field, then parents will only be able to select dates not yet claimed.