You can add the built-in "Partner Type" field to your Become a Partner form to allow prospects to specify their desired type of Partnership when applying to join your program.
Become a Partner Form
To add the Partner Type field to the Become a Partner form, go to Settings > User & Organization Fields > Organization Field Requirement & Visibility. You'll then select the third checkbox (on the right ) under the "Become a Partner Form" column to make the field visible on the form. 
Organization Field Requirement & Visibility Settings
You can select both checkboxes under the column to make the field visible and required.
Organization Field Requirement & Visibility Settings
FAQs:
Can Partner Users update their Organization's Partner Type in Channeltivity?
Partner Users do not have the ability to update the built-in Partner Type field in their Organization Profile as it is a read-only field to Partners.
Organization Profile - Partner User View
Related Documentation:
What is the difference between "Become a Partner" and "Request Access" forms?