How a narcissist parent can affect a child, the child may not feel heard or seen, they might feel like there own reality and feelings are not valid, they may feel like the other parent treats them as an accessory and not a person, the child might be taught how they look is more important than how they feel. They might be taught to keep secrets from you.
The child will be more valued for what they do for the parent rather than for who they are as an individual . They may start to feel used and manipulated. They will try and seek approval from the other parent. They might start to not trust people. They may not learn routine or boundaries.
They may feel criticized and judged, instead of accepted and loved.
The child will not learn appropriate boundaries for relationships.
They can often feel shamed and humiliated by a the other parent.
This can all be counterbalanced with one healthy loving parent. Limited contact with the toxic parent. If you have routine and boundaries in place the child learns for themselves that this is a good environment to be around. They child will learn for themselves what kind of people they like to be around. They can grow up happy and healthy knowing what kind of people to avoid. They will probably always visit the other parent but the will grow to limit it themselves. The healthy parent can provide the tools for a health childhoods. Look at the post about the love languages of children. One parent can provide all these things. Not saying it easy but it can be done
you will have double duty as the responsible one. You can ward off the affects,Simplest approach is to parent with empathy the antithesis of narcissism. If you can get the children in with a good support network and are and help them learn assertiveness skills to use with a parent who does not emotionally validate them, speak to health visitors, Docters. Make sure you are heard. Some will have seen it before. Make sure you keep a diary of any thought your child is saying to you that seems inappropriate for the age.