In reply to DaveHK:
The majority of the main summits are good scrambles/mountaineering days: Torrecerredo, Torre del Llambrión, Tesorero (Central Massif); Torre de Santa María (Western Massif)... to name just a few. They all are grade I-II scrambles, plus long approaches; big days in the mountains as a whole.
Upping the game a bit, there are many classic ridge traverses: Argaos (Western Massif), Cabrones-Torrecerredo (Central Massif)... but whilst you may not need to pitch any section of the climb, it is standard to do them with a harness and rope to abseil sections that otherwise would be steep, exposed downclimbs.
For a solo trip like yours (no climbing gear, no partner), taking a harness and perhaps 30 m. of cord might be a good compromise. My overall impression is that with a detailed knowledge of the area there is loads that you can do with no gear whatsoever, but being able to abseil (if only short sections) opens many possibilities, plus added reassurance in terms of route finding problems etc.
The guidebooks buxtoncoffeelover refers to are, both by www.desnivel.com:
Escaladas Fáciles en Picos de Europa - Fuente De
Escaladas Fáciles en Picos de Europa - Vegarredonda