Many children realize at a very young age that they can "play" their parents against one another to get what they want. When parents are separated or divorced, this can be even easier to do. Often, all it takes is to tell one parent that the other won't let them do (or buy) something to get the other parent to agree. Divorced couples are often vying to be the best-loved co-parent, and what better way than to give your child something your ex won't? It's also a good way to get under an ex's skin.
Kids may be able to get what they want at the moment this way. However, in the long run, they lose the security of knowing that their parents, despite their differences, are united in parenting them.
That's why it's important for co-parents to realize when they're being pitted against each other by their kids. This can help you take steps to prevent it and make important parenting decisions together (assuming that you share custody of your children).
Kids will often approach parents separately with a request. If they don't get the response they want from one parent, they go to the other. Co-parents often make decisions on their own for small things, since they're living in two separate homes. However, when it comes to things like a new phone, going to their first PG-13 movie or spending the weekend with a friend and their family, it may be best to have the permission of both parents.
If they ask you for something without their other parent around, tell them that the two of you need to discuss it and get back to them. You're not asking your co-parent's permission. You're simply discussing it together and agreeing on a decision. Fortunately, this is easy to do via text, Skype and numerous other methods of instant communication no matter how far apart you are.
If you're having a serious issue with your co-parent making decisions that you believe are harmful to your children's well-being, and you aren't able to resolve the problem by talking with them, you may want to consider seeking additional provisions in your custody agreement regarding discipline and decision-making. Your attorney can help you work to do that.