The rules are somewhat different depending on the plan your parent has.
If your parent is covered under a health plan offered by an employer with 50 or fewer workers (“small group” plan) or through an individually purchased ACA-compliant health plan on or outside of the Marketplace, then the plan is required to cover your prenatal care and delivery. Many prenatal services are covered without cost sharing, but labor and delivery may be subject to cost sharing.
However, if your parent is covered under a health plan offered by an employer with more than 50 or more workers (“large group” plan), then the plan is only required to cover your prenatal care, but is not required to cover the delivery, or may charge cost sharing. Medicaid covers prenatal and delivery services in all states. You may be able to qualify for Medicaid on your own depending on your income.
Your parent’s plan, regardless of the source, is generally not required to cover your child as a dependent. You will need to obtain coverage for your baby. Depending on your income, your child may be eligible for coverage under the Medicaid/CHIP program in your state. Or you can buy a family policy through the Marketplace and, depending on your income, you may be eligible for a premium tax credit and/or cost sharing subsidy to reduce your cost of that coverage.