Just because Dolly's not willing to cede her own power to her grandparents doesn't mean she's not perfectly happy to partake in the humiliation of her mother for the sake of making old people happy. But there's a lot more going on here than just another one of Dolly's power plays.
On a superficial level, this cartoon is a simple mother-in-law joke. If you don't immediately recognize it as such, it's because it's told from the daughter-in-law's point of view, which is a fair amount rarer than the mother-in-law joke told from the son-in-law's point of view. Underlying the latter variety is just hack misogyny: "Aren't women annoying! Ha ha!" But underlying the former kind of mother-in-law joke is a more complex social critique, which this particular Family Circus cartoon really brings to the fore.
On a basic level, these jokes are about generational conflicts and what should be expected of women within the family and culture more broadly. The cartoon above lays out the dichotomy pretty distinctly. In the foreground, we have the new generation, while in the background, we have the old generation. The new generation wears pants; the old generation wears a dress. The new generation looks harried, despite only being responsible for a single dish; the old generation looks serene, despite having a whole table in front of her. The new generation struggles to control her child; the old generation is in control of the entire table, a symbol for the whole family. And as per the dialog, the new generation often abdicates her wifely duty of cooking dinner for her family every night and forces her husband to perform menial tasks for her, like pick up food from a restaurant; the old generation never does, not even ONCE!*
*(My favorite touch is easily Grandpa standing off to the side reading the paper, just like it's supposed to be. The old generation would certainly never think or dare to interrupt her spouse's hard-earned leisure time.)
This trope generally sides with the younger generation, acknowledging that the world is different now and that the older generation never actually found everything as easy as it tends to claim in the first place. I don't think that's the case here. On the contrary, it seems pretty clear that, despite not being entirely unsympathetic to the younger generation, Keane Inc. is still on the older generation's side. Which, of course, is exactly what should be expected.
The most bizarre thing about all of this is that Thel Keane is the stand-in for the younger generation. Would that be the same Thel Keane who has never had a career of any kind, is generally held up as the perfect stay-at-home mom, and pumps out babies seemingly every year? Yep, that would be the one.