Can you run an additional breaker panel parallel to the existing one?

It is not in parallel. You put a breaker in existing panel and use that to feed the subpanel. The Amperage of this subfeed breaker should match the rating of the new panel. For example a 100A breaker might be typical. Remember that ground and neutral are only "bonded" at the main panel. Usually a subpanel has a means to separate the neutral and ground in a subpanel. Be careful since everything about doing this is dangerous.

Another Answer

Yes, you can have two breaker panels in parallel. If you had a 100 amp panel on a 200 amp service (or increase the size of the service, check with your power company on the size of your service, you could add a second 100 amp panel in parallel with the first. You can have up to 6 disconnects per service, but they must be located adjacent to each other or in the same enclosure. So either install the second panel beside the first or a 100 amp disconnect beside the first panel and feed out of the disconnect to the new panel located where you need it.