The primary purpose of life insurance is to prevent loved ones from hardships related to your death. In a situation where the parents have separate homes, this could be especially important, because the responsibilities towards the children are shared differently between the parents. For example, one parent may have custodial responsibility, while the other has a financial obligation. This is exactly the kind of situation where a permanent life insurance policy could have the most benefit.
You cannot take out a life insurance policy on your child's other parent without his or her written consent. Typically, when you apply for the policy, the other parent will have to take a medical exam, and then sign off on the policy. Without their express permission, the only way you could take out a life insurance policy on them is under the direction of the court.
Think of the benefit to your child as being compensation for the loss of a parent. An insurance policy cannot bring the parent back, but it can pay for the things that the other parent should have provided. In some situations where child support is a contested issue, the child could actually end of benefiting more from the life insurance policy than from the parental contributions themselves.
Your child needs you to have an insurance policy on yourself, as well. When both parents are properly insured, the children are assured of living at their accustomed standards even if one or both parents should perish. It is not enough for one parent to be insured, because disaster can strike from any direction.
Assigning one or both policies to a trust fund may be the best option. By naming a trust as the beneficiary, you can avoid having to pay estate taxes out of the policy proceeds, as well as defining how the money is divided out as time goes one. For example, a trust can be established to pay the regular monthly bills for X number of years, and then pay the remaining balance to the child.