The official Antena 3 website (antena3.com) offers many full episodes for free, supported by ads. However, they usually rotate the catalog. Older episodes tend to be removed to make space for new ones. If you are looking for capítulos completos gratis from the last two seasons, this is your best official bet.
The top playlists usually have titles like:
A hidden gem. Fans have uploaded entire seasons of El Secreto de Puente Viejo to the Internet Archive. Search for "El Secreto de Puente Viejo Complete Season 1." These are download links (MP4), but completely legal under fair use? The legality is gray, but the archive is non-profit and rarely removes fan-uploaded classic TV.