I don't know anything about Balsam, but this message has been on the Hybridizing Forum for two months so here is a little information.

This page has a link to a 1958 article in PDF format on hybridizing Impatiens balsamina to get various floral colors.

It looks like balsam flower color genetics can be fairly complicated. The table lists parent colors in the first column, the F1 color in the second column, genes in the third colum, and F2 colors in the fourth column. In crosses numbered 21 and 22 they got purple by crossing pink with either lavender or nearly-white lavender.

Color descriptions may vary so you may need to experiment to compare to their descriptions.

It would perhaps be easier to buy some seeds and make a selection, most mixtures include purple.