Equity does not represent production divisions in a company (i.e. chocolate, strawberry, and vanilla does not make sense).
In Sole proprietorship, equity represents 1 owner.
In Partnership, equity has at least two sub-accounts, namely Partner 1 and Partner 2.
In Corporations, equity may have Common Stockholders and Preferred Stockholders, or even different class of shares for insiders and angel investors.
As you can see, equity represents who owns the company, and how much they own. It is not what the company does or manufactures.
First and foremost, define the boundary of the firm. Are your books titled "The books of the family of Doe", "The books of Mr & Mrs Doe", or "The books of Mr & Mrs Doe & Sons".
Ask yourself, who "owns" this family. If you believe that a marriage is perpetual until further notice then it does not make any sense to constantly calculate which parent owns the family more. In partnership, firm profits are attributed to partner's accounts using previously agreed ratio. For example, (60%/40% because Partner 1 is more hard working and valuable to the firm.
Does your child own this family? Does he/she have any rights to use the assets, to earn income from the assets, to transfer the assets to others, or to enforce private property rights? If they don't have a part of these rights, they are certainly NOT part of Equity.
So what happens to the expenses of children if you follow the "partnership" model? There are two ways. The first way is to attribute the Loss to the parents/family since you do not expect the children to repay. It is an unrecoverable loss written off. The second way is to create a Debtor(Asset) account to aggregate all child expense, then create a separate book called "The books Children 1", and classify the expense in that separate book.
I advise using "The family of Doe" as the firm's boundary, and having 1 Equity account to simplify everything. It is ultimately up to you to decide the boundaries.