Depending on the single parent's personality, mental health and how they have conducted themselves post divorce in the co-parenting arena (with respect to the children, the former spouse and the step-parent), they may not be willing to "accept" that their children have a new family, in addition to the one they share with the single parent. In this scenario, the single parent may feel excluded, angry, envious, sad, insecure, etc. Such emotions may be exacerbated by mental health disorders such as anxiety, depression and personality tendencies/disorders (e.g. Narcissistic). Comorbidity is not uncommon with such disorders. Further, the single parent's negative feelings may escalate due to their life not turning out how they had anticipated. For example, such person may be lacking a "significant other" in their life which perhaps was not what they had planned, especially as they age.
Depending on the severity of such emotions, the single parent may "lash out" at the children's other parent and step-parent, including intense efforts (involving fabricating statements that are untrue) to manipulate the children in an attempt to interfere with the relationship between the other parent and step-parent. Sadly, the children become "pawns" that the single parent uses to create conflict or alienate his/her biological children from the other parent and step-parent. Perhaps the single parent claims to be a victim, and attempts to instill feelings of guilt in the children whenever they wish to spend time with his/her other parents (including grandparents) with hopes of preventing them from having a healthy and loving relationship with their other parent and step-parent. The single parent remains self-focused on his/her own perceived personal needs at the expense of what's in the best interest of the children.
Based on my education, professional experience and on-going review of the literature, I believe the behaviors of the single parent described above are psychologically damaging to the children and can have long-term affects on the children's mental well being. Including, impacting the children's ability to have their own healthy relationships with others in the future. If you find yourself acting in a manner that is similar to that which is attributed to the single parent above, I encourage you to seek professional help via a therapist to assist you in dealing with your emotional struggles. At no time should a parent attempt to utilize a child as a "therapist" by discussing their emotional struggles and negative feelings toward the other parents. As an adult, such discussions should take place privately with another adult, preferably a personal therapist.
When all parents (biological and step-parents) of the children stay focused on what's truly best for the children, one should realize that it is equally important for the children to have healthy loving relationships with all of their parents.
To ease the emotional burden, related stress and internal conflict children may feel (especially when the above scenario is at play) in this type of situation, I recommend that both biological parents and the step-parent(s) inform the children, that it is okay for them to care about and love their new step-parent. Explaining that their step-parent is not in place of their biological Father/Mother but rather in addition too. It is another person in the childrens' lives who will love, care for and facilitate the childrens' well-being in this complicated world. Post conversation, parents should be committed to ensuring that their behaviors/comments are consistent with the above message. As adults, we realize having the above conversation with the children is not dependent on whether you are "friends" with the other parents (biological and step-parent). Such friendship or lack thereof is irrelevant to this issue.
I realize it takes a secure and emotionally healthy adult to be able to have the aforementioned conversation with the children and display consistent behaviors thereafter. However, I strongly encourage parents to do so for the sake of their children.