Parenting is a delicate dance where the adults in the household must maintain unquestioned control over the other residents (i.e., the children). Some spouses may have different parenting techniques they like to pull out when they're the sole caregiver. That's fine. It's important, though, to understand compromise as adults, and to continually present a unified front for the children. When toddlers and preschoolers see discrepancies between parents, they will use those to their advantage, even at this young age. A parent that undermines his partner may find himself in line for spoiled kids later on.
Wheedling and Whining
When children realize that one parent will always side with them against the other one, all they have to do is ask one in front of the other for something. The mother could say no for hours on end, but all the daughter has to do is wait until her father gets home to ask him. This is made worse if the mother tells the father about the ordeal and asks him to stand by her decision. Allowing something that has been disallowed by the other parent may make you the good guy in the short term, but it teaches children they don't need to listen to the adult figures in their lives.
Whining in the long term eventually takes the leap into manipulation, where toddlers and preschoolers throw tantrums over what they want, knowing full well that if one of the parents can withstand the crying and noise, the other will cave. This shows the kids that each parent has a limit and that the children can surpass that limit with ease. All they have to do is be loud enough and annoying enough. If the parents don't stand together with consistent discipline for these actions, the children learn that if they just keep up long enough, they will win. As they get older, the tantrums will turn to lying and emotional manipulation with the aim of getting what they want.
Inability to Handle Emotion
Manipulative children, or children that have been given the control in the household, due to one parent constantly undermining the other, will have trouble handling intense emotions. They won't understand patience, delayed gratification or sacrifice. This will show in their daily dealings with other kids and parents, and they will suffer socially, particularly if they use sneaky or lying techniques on those outside the family to get their way. Young children are shown that adult authority is meaningless, leading them to believe if they only ask more persistently, they can get what they want.
Lacking Respect for Authority
As small children progress into young adulthood later in life, they can have trouble with important authority figures like teachers, police officers and any adults who need to guide them through life. If they don't grow up respecting their parents and understanding that rules are in place for a reason, they will carry that over to other authority figures as they grow. They will be looking for strong boundaries, and pushing hard against them. When one parent undermines the other, the children learn that what adults say doesn't matter.