If you have recently been married it is most likely that you and your spouse no longer lives with your parents. If this is the case then normally the parent's insurance would not want to cover you any longer since you now live with your spouse in a separate resident and not in the same household as any parents.
Each state and insurance company has their own rules and guidelines regarding auto insurance so you, your spouse and the parents involved will need to find out for certain about if either of you can be kept on their insurance policy.
When you get married and move out on your own with your spouse you normally will both need to obtain a car insurance policy to cover your vehicles together in both of your names. If you own multiple cars then you may be able to get a multi-car discount.
If you would like to start shopping around for car insurance online for you and your spouse start here and now.