What part time jobs offer health insurance?

UPS does. However, the catch is that you must work a minimum number of days/hours to qualify for benefits each month. The problem is that everything with UPS is based on union seniority. If there aren't enough hours to go around, a more senior employee can bump a new hire out of their shifts so it's not uncommon for a new employee to not have health insurance during much of their first year because they are only working a few hours each week.

That's totally optional on the specific employer's part. There are no rules for specific categories of jobs that will so provide.