The family member sets up the skill in their time zone and with their availability hours using the Alexa account. This information is stored in DynamoDB and recalled when the parent invokes the skill. The skill invocation can be simplified through the use of an Alexa Routine to start the My Calls Skill
Instead of Alexa, open My Calls" they could then say "Alexa, is this a good time to call Jeff? If it's not a good time for the call, the parent is offered a Reminder to call If it is a good time, the Skill exits after telling the parent to say "Alexa, call Jeff" as Alexa Skills can't place Alexa Calls