There is no set age at which a child can decide if they wish to visit with a parent, except that once the child is 18 years of age, s/he is no longer considered a minor and visitation orders do not apply to the child (absent special circumstances such as a guardianship for a disabled child, etc.). That being said, courts can use their discretion to determine how to enter and enforce visitation orders based on the expressed preferences of the child, particularly older children. However, that varies and is determined strictly by the specific facts of that case and not by any general policies or laws.
The response given is not intended to create, nor does it create an attorney-client relationship or an ongoing duty to respond to questions. This is not intended to be anything other than the educated opinion of the author. It should not be relied upon as legal advice. The response given is based upon the limited facts provided by the person asking the question. To the extent additional or different facts exist, the response might possibly change. For a specific answer to your question, you should speak with an attorney who practices in this specific area of law.
The judge decides matters of visitation, not the child. Yes, at age 14, children will be "heard" in court as to their preferences, but that will not control the outcome. The best interests of the child is the guiding principle for the judge. All too often, feuding parents use the child as a pawn, so judges will look carefully at the reasons for the child's refusal.
Is the parent abusive, neglectful? The court will use these measures to consider teh child's wishes.
Barring some type of significant issue with a parent, such as abuse for example, keep in mind that children do best when they have good relationships with BOTH parents. Although your child may not want to visit with one parent, it could be more beneficial in the long run to encourage the 14 year old to spend time with the "other parent."
Also, I would suggest you encourage the child and perhaps both parents (not necessarily together) to meet with an experienced mental health professional in an attempt to strengthen the bond between the child and both parents.
Good luck, and remember that without knowing all of the specifics, this info is just provided as general info, and not specific legal advice. You should consider meeting with a lawyer.