This happens when Caregiver A says to Billy or Sally, "Don't listen to your other parent. Here's what we'll do." It can sound fun and conspiratorial: "Your mom's away. Now us boys can have fun," or outright damning: "Your other parent's a jerk. Don't listen to him!"
Kids need every opportunity to build and maintain the healthiest relationship that they are able to with each of their caregivers, regardless of the adults' feelings for one another. Unfortunately, human beings become physically mature enough to have kids long before they become emotionally mature enough to put their own feelings aside in support of the child's needs.
2. Enlisting your child as a spy, courier or co-conspirator.
When a parent begins to treat a child like a peer (a process known as adultification), the child feels complimented and eagerly accepts the new role. It feels like a promotion! But adultification cheats kids out of the opportunity to feel safe and cared for. It inevitably exposes them to experiences that they're not ready to cope with emotionally. The result is a huge amount of anger and anxiety, depression and distraction.
Don't ask your child to quiz their other parent when they're together, or report back to you afterwards. Don't let your own loneliness, neediness or stress make your child into a new best friend, or co-parent to a younger sibling (a process known as parentification). Get your emotional gas tank filled by other adults. Parenting means you refueling your kids, not the other way around.
You should give "we" messages. "We" communicates that the co-parents are mutually participating in a decision despite their differences, as in, "Last week you asked me if you could go to the dance. Your dad and I discussed it and WE think it's okay." Aside from the obvious content (go to the dance), WE communicates that the parenting team is always together, creating a safety net that communicates security to the child.
3. Trying to do it all alone.
Raising a child does, indeed, take a village. To make sure your emotional needs are met (not to mention getting some time away), it's critically important for you to have suitable co-caregivers. That doesn't mean getting married, necessarily; it sure doesn't mean staying married if you're in an unhealthy relationship. It means finding healthy support for you, and assembling a team to raise your child. Your adult partner could be a co-parent; maybe your sibling, your mom or dad, or your neighbor. Their pediatrician, clergy person, and a good child therapist can all be part of the team.