It would not likely be more appealing for them. Consider that even if they don’t have to pay you, they are still relying on you to help advance their research, and doing a part-time PhD means that their (your) research is going to be moving at a glacial pace. Meanwhile, their competitors in the field are moving forward with full-time students at a normal pace and very well might outpace them and beat them to the punch on your topic.That translates to a lot of wasted time. Further, many grants have time limits on them when sponsored by industry or organizations like the Air Force who want to use the results.
Essentially, part-time PhDs are not common because doing a PhD is not a part-time job.