Paris to Pamplona by TrainThere is no direct train from Paris to Pamplona, but there is good service from Paris to Vitoria, though this leaves you in Vitoria at a ungodly hour of the morning. You can then take a bus from Vitoria to Pamplona.
The return journey from Vitoria to Pamplona is much more reasonable, as it leaves you in Paris at a sensible hour of the morning.
An alternative is to take a train from Paris to Hendaye, walk across the border to Irun and take a bus to Pamplona. However, there is only one bus per day from Irun to Pamplona. A sensible option is to instead go from Hendaye to Bilbao, spend a night there and then travel from Bilbao to Pamplona the following day.
Paris to Pamplona by Bus
There are buses from Paris to Pamplona, but it is incredibly difficult to book them online. The Spanish bus timetable site, Movelia, allows you to book tickets from Pamplona to Paris (departing on Monday, Wednesday or Saturday) but not for the return journey. The company they say runs the service, Conda, doesn't offer tickets for either direction on their website.
Eurolines also offers infrequent buses from Paris to Pamplona.
Paris to Pamplona by CarThe 900km journey from Paris to Pamplona takes about eight hours by car. Follow the A10, A63, N10, A63, A8 and A15 roads. Note that some of these are toll roads.
Read more about Renting a Car in Spain.