It looks as if howtheduck is right when he says that Elly is the go-to Foob for baby-sitting.
Panel 1: That's because Deanna is singing Flapandhonk's praises. She cannot, she tells Liz, imagine what she'd do without them.
Panel 2: This is because, as a Patterson male, Mike's primary focus is his career. It matters not that he works out of his home, he must be kept isolated from his children for that is the Code of the Foob. Since she must work as well, that leaves them with the one option.
Panel 3: They'd like to be independent but they need the help. Liz can foresee no problem with Elly sitting for her too. She really isn't very good at predicting the future, is she?
Panel 4: That's because we see her after having asked Elly. Flapandhonk thought bubbles that she's already raised HER children.
Summary: That's the thing about making your children hopelessly dependent on you; it makes you lose your own independence.
Prediction: Cue the flashbacks as Elly once again Looks Back In Martyrdom.