Is it possible to have multiple versions of one form?

Is it possible to create a master form that can then be parsed out to different pages with certain fields or values hidden?  Ultimately I would love if those "sub versions" could be on unique web pages but add all submissions to one master database record

It is not possible to have multiple versions of the same form, I mean you cannot have more than one form and have  all the submissions in the same form. 

What I would suggest you is to create one form only with all the fields you need, then add the Get Form Page URL widget that will help you to know the webpage where the form is embedded this way you will be able to show certain fields based on this. 

Here's the link to the widget: https://widgets.jotform.com/widget/get_form_page_url 

This guide will help you to add a widget to your form: How-to-Add-a-Widget-to-your-Form

This guide will help you to set up the conditions to hide fields based on the link where the form is embedded on: How-to-Show-or-Hide-Fields-Base-on-User-s-Answer

Hope this helps.