Fly from Paris to Madrid with the cheapest airline available on your travel date (tip: don't overlook Easyjet). Once in Madrid, Alsa has buses from Madrid airport to Astorga (or, if you prefer, to Ponferrada). They have more buses from downtown Madrid (Estación Sur de autobuses) if the schedules of those out of the airport don't suit you. An alternative from downtown Madrid would be to take a train out of Chamartín station (either to Astorga or Ponferrada) but train service is much less frequent than bus one.
Alternatively, Vueling has some dates (it isn't daily) non-stop flights between Paris and Asturias airport. That would be an option to consider if you wanted to make part of the Norte or the Primitivo. Flight to Asturias airport, bus from Asturias airport to Oviedo and bus from Oviedo to Tineo to start in Tineo your Primitivo walk. Or bus from Asturias airpot to Avilés and bus from Avilés to Luarca to start in Luarca your Norte walk. All these bus lines are operated by Alsa.
If you prefer to go to Puebla de Sanabria (on the Sanabrés) or to Zamora, once in Madrid go from the airport to Estación Sur de Autobuses (on the downtown) and take a bus from Madrid to Puebla de Sanabria or to Zamora with Avanzabus. Alternatively, you can take a train out of Chamartín station to either Zamora or Puebla de Sanabria.
To commute from the airport to Estación Sur de Autobuses, you can take a cercanías train from Terminal 4 to Méndez Álvaro train station. I you arrive to other terminal, take a (free) airport shuttle bus to terminal 4. The metro (subway) would be another alternative but it takes much longer and involves minimum 1 change (from line 8 to line 6) although it might be quicker to make the route with 2 changes...
To commute from the airport to Chamartín station, take a cercanías train from Terminal 4 to Chamartín station. If you arrive to other terminal, take a (free) shuttle bus to T4.
Links of the bus companies and the train one to look for info about schedules and fares (note: it's early to book train tickets for June and it may be to book some bus tickets):
http://www.alsa.es
http://www.avanzabus.com
http://www.renfe.com