No, large employers are not required to offer health benefits to part time employees and there is no penalty for large employers that don’t offer health benefits to part-time employees or their dependents. If you work part-time and you are not offered health benefits, you (and your family) can apply for coverage in the Marketplace; and, if your income is at least 100% of the federal poverty level, you can apply for a premium tax credit that may reduce the cost of coverage in the Marketplace.
Note that a part-time employee is one that works, on average, fewer than 30 hours per week. If your hours vary during the year, your employer may have some options in determining your status as a full-time or part-time worker. Your employer can tell you whether you are a full or part-time worker.