Spain's national parks are run through MITECO (the Ministry for Ecological Transition), and its own official visitor guidance for parks like Ordesa y Monte Perdido lists 'llevar perros sueltos' — bringing loose dogs — among the prohibited activities inside the park. The guidance doesn't describe a zoned core-area outright ban the way several French national parks do; instead, the rule targets an unleashed dog specifically, which in practice means a dog kept on a lead is the way to visit with your own pet.
This is a meaningfully different, more relaxed starting point than the France section of this guide covers, where several major parks ban dogs entirely from their core zones even on a lead. It's still worth checking the specific park you're visiting before a trip, since individual parks can set their own additional detail on top of the general loose-dog prohibition, and some sensitive wildlife areas or specific trails may carry extra restrictions not covered by the general park-wide guidance.
The underlying reasoning is the same one that applies anywhere dogs share space with protected wildlife and livestock: even a calm, well-trained dog off the lead can disturb nesting birds or provoke a reaction from a working livestock guardian dog in the same terrain. Keeping your own dog leashed throughout a Spanish national park visit is the straightforward way to stay compliant and avoid an avoidable wildlife or livestock conflict.