Prison vans are parked on roadsides for hours outside after dropping off defendants, while garda vehicles also line the streets.
The Irish Prison Service has said that, while its vehicles can get into the car park, there is not enough room for them to stay.
The Courts Service has defended the building -- which was officially opened by President Mary McAleese on Saturday. The service insists it is up to the drivers of all "non-operational" vehicles to find their own parking in the area.
It is holding talks with the Prison Service and Dublin City Council to iron out the problem, which has escalated since more legal business was moved to the CCJ at the start of the year.
Members of the public have complained of vehicles visiting the Parkgate Street complex parking on surrounding roads.
"Our vehicles have no problem accessing the building," an Irish Prison Service spokesman said. "All our vans go inside for prisoners to disembark.
"Some of them come outside and park there because there is a capacity issue inside. Issues around parking for vehicles outside the courts are being discussed between us, the courts service and the city council with a view to resolving them in the short term."
The Courts Service defended provision of parking within the building, saying the space provided is exactly what was agreed with the Prison Service.