Ordinarily, you don't have the right to deny the other parent their visitation rights. However, there are exceptional cases in which you may block the other parent's access to their child, at least temporarily. Here are four examples of such cases:
If There Are Preexisting Engagements
If the parent shows up without notice or prior arrangement, then you don't have to cancel the child's plans or your plans and let the other parent have them. This is especially true if the preexisting arrangement is something that is beneficial to the child's welfare. Consider an example where the child is preparing to attend a school play and the other parent shows up (unannounced) to pick up the child; in this case, you can refuse the visitation.
If The Other Parent Is Intoxicated
Driving while intoxicated is dangerous for all road users, including passengers. You should never let your child get into a car with an intoxicated driver even if the driver is the child's other parent. Therefore, if the parent is intoxicated and they want to pick up the child, you are within your rights to refuse the visitation and tell them to come back when sober (assuming they insist on driving).
If The Other Parents Wants To Take the Child to a Dangerous Area
There are areas where children should not be and even the government has a say in the matter. For example, minors are not allowed to go to nightclubs or bars where alcohol is served. Therefore, if you suspect that is what the other parent wants to do, you have the right (and should) not to agree to it.
If You Suspect Child Abuse
Lastly, you should also deny visitation if you suspect the other parent of abusing the child (for example, in past visits). Child abuse is a serious criminal offense that you should never allow to happen if you can do anything about it. Remember that child abuse comes in different ways; it can be emotional, physical, and sexual or take many other forms. For example, if you suspect that the other parent won't provide the child with food when they are together, you have the right to stop the pickup.
Note that you should document all these cases to ensure you can prove them if the other parent accuses you of denying them visitation. Also, it's best to consult the court after the denial, especially if you think it will reoccur; for example, it may be advisable to change the visitation rights. Consult a divorce law office to help you with the way forward.