I assume that you and your boyfriend both signed the lease together and your parents signed as guarantors. The landlord will go after the remainder of the lease along the path of least resistance: first you and if that fails then your parents.
If either of you (you or your parents) end up paying your boyfriends share of the rent, then you can sue him fairly easily in small claims court. You don’t need an attorney, the cost of filing is inexpensive, and it is procedurally simple. Bring a copy of your lease and a receipt for the amount paid.
This is not legal advice and is not intended to create an attorney-client relationship. You should speak to an attorney for further information.
Unfortunately, all parties who signed the lease will likely be on the hook for the rent due. The upside is that if you break your lease, your landlord will be obligated to mitigate or reduce their damages by attempting to rent the apartment as quickly as possible. The downside is that if you stay on during the term of the lease, you'll likely be jointly and severally responsible for the rent due, which means you cannot just pay your share of the rent, but will have to pay the full amount due. If you don't, then your landlord can proceed with an eviction and obtain a judgment against you and your boyfriend for the amount owed and removal of you from the apartment. This is exactly the type of scenario landlords try to protect against by having guarantors co-sign rental agreements. You may want to have an attorney review your rental agreements to determine exactly what kind of liability your parents signed up for.
This "answer" is for information purposes only and is not intended as legal advice or to create an attorney-client relationship.
The two previous answers are great so with my answer I will sum it up for you.
1. Everyone who signed the lease is on the hook for the rent payment to the landlord.
2. If your ex-boyfriend doesn't pay his share then you may have a claim against him personally.
This information does not constitute legal advice. Without more information this is merely an answer to a question. If you would like legal advice tailored to your specific questions you should contact and retain a lawyer.