Are you working in Lightning or Classic? You should be able to use an Action to accomplish this. For example, if you wanted to create an Opportunity (parent) from a Quote (child), you would create a new Action on the Quote record.
You would then drag any fields you want the user to fill out onto the Action's page layout.
Then you can also set predefined field values as part of the Action. This example shows the Opportunity Close Date as 5 days from today.
Then just drag the button for the action onto the child record's page layout. If you are in Classic, the only thing is that this Action will be present in the Chatter feed for the record, not at the top with the standard buttons. You'll add it to Quick Actions in the Salesforce Classic Publisher. In Lightning, you'll add it to Salesforce Mobile and Lightning Experience Actions, and it will appear in the upper right corner with the rest of the buttons.
Here's a link to the SF Trailhead for more details: https://trailhead.salesforce.com/en/modules/salesforce1_mobile_app/units/salesforce1_mobile_app_actions_objectspecific