Anon NYC has specially requested that we address this particular issue.
Françoise shows up on Monday in her pyjamas. It is unknown whether or not she had been in bed before, but the sneaking around the corner and her eventual destination by Thursday’s strip sort of implies she was in bed. So the situation would be that Liz and Anthony and Françoise had dinner, put Françoise to bed, and then were preparing to read over Mike’s new book together. We see toys out on the floor, so it appears that Anthony is not strict with Françoise about putting her toys up.
Then when Françoise arrives saying she wants to see Mike’s book, Anthony enthusiastically tell her she can see Mike’s book. He doesn’t say, “Go back to bed.” So, it appears that Anthony is not one to restrict the things Françoise is or is not allowed to see. Liz, on the other hand says nothing, but she does lean away from Anthony when she realized Françoise is there, as if to say she considers physical contact with Anthony in front of Françoise to be inappropriate.
Without anything to lead into it, Françoise is between Anthony and Liz and saying, “My daddy!” Despite this clear attack, Liz takes it lightly and by deferring to Françoise as to where she sits, actually seems to be giving her the authority to do what she just did. So, these are two things Françoise has asked for which she gets from the adults. Anthony is showing her the book and Liz gets and leaves Françoise alone with her father. These actions cry out for one of these two to put their foot down to this girl.
Anthony makes a move to discipline by telling her Elizabeth was sitting there first, but Françoise completely rejects the discipline to Anthony’s face. This is what I would call backtalk in my house. Anthony caves in, and instead of discipline, tries to reason with her.
Elizabeth reacts to Françoise by leaving Anthony to explain things to her. As I mentioned before, this would have been a perfect time for Liz to say, “I am not trying to take your father from you.” Instead she is leaving all the discipline and all the punishment and all the soothing conversations to Anthony. What is presented then is the idea that Liz Patterson is leaving the parenting to Anthony, and this is appropriate because she is the girlfriend and not the fiancée yet. I agree that Elizabeth should not usurp Anthony’s parenting, but if there were parenting which needed to be usurped, it is this stuff. Anthony as a parent is trying to reason with little Françoise, which is not going to work because the fears she has about Liz are not rational thought.
After this, Françoise heads off to bed (presumably Anthony tells her to, but we don't get to see it), but then she doesn’t get very far before she is back. We don’t see Anthony, the parent or Elizabeth try to tuck her in bed, so it is not too surprising she is back out again. Anthony blames Françoise’s behaviour on her inability to accept another woman in the house. Liz, on the other hand, plans to do nothing about the situation and assumes Françoise will eventually come to her. Somehow Françoise gets in Anthony’s arms and shoots Liz a dirty look. Françoise is running all over both of them, and they don't even seem to be trying to stop her.
In Thursday’s For Better or For Worse, once again both Anthony and Liz stop what they were doing to deal with Françoise’s demands.
Anthony as father: Gives into Françoise at every turn. The only hint of any parental discipline is when he tells Françoise she is not being nice. The only hint of understanding is when he tells Françoise she can’t substitute for Elizabeth, because he wants a big girl. Why he can't explain to Françoise that he likes Elizabeth, I don't know. It makes it seem like he only likes Elizabeth because she is a grownup.
My viewpoint: Just awful parenting. The kid is running all over him, so he can’t enjoy his time with Elizabeth. He is not even trying to deal with her overpossessiveness.
Elizabeth as girlfriend, since she is not trying to be mother: Her only interaction with Françoise was to ask her if she could still sit by Anthony, which is in deference to her rudeness, which Liz does not appear to be taking seriously. Liz’s overall plan seems to be to leave the parenting to Anthony and wait for Françoise to come to her. In other words, she is taking the laziest route possible. There is no action there at all, except inaction. I can't see how her plan will work, but I know it will.
In any case, these two are a long way from getting Françoise to call Elizabeth "mommy" instead of Thérèse.