There is not one button that will do this but there is a work around.  I also have partnerships with a lot of partners and multiple states.  Here is what I do (it's not perfect but it's better than the alternatives.) Click the print icon.  Check the federal and state boxes and then in the lower left click on "choose items to print."  From there you can expand all folders and choose the forms that pertain to just the partner you want to print.  I uncheck all and then choose the K-1s and K-1s for one partner.  Same for states.  You will have to do that for each partner.  You will end up with a file for each partner with just the K-1 forms that you choose.