Is it permissible to lie to the mother in order to go to the father if she refuses, and is obeying the mother by not visiting the father considered disobedience to the father?

It is not permissible for parents to incite their child to disobey one of them. The child should not respond to such incitement and should gently explain its impermissibility to them. If they persist, the child may use equivocation (tawriyah), allusions (ma'areed), or even lie if necessary, to make each parent believe they are content with the child's relationship with the other, while also being able to visit one of the parents secretly.