In October 2017 I visited Talampaya National Park by public transport from La Rioja, which is much easier than I was led to believe by the information I read on various websites. I took the 7:00 am bus operated by Facundo from La Rioja, destination Villa Unión, which travels through Talampaya National Park. Around three hours from La Rioja, the bus passes the park's main entrance, where the bus will stop upon request (I was the only passenger to disembark at the park, so don't rely on others to request the stop). The bus arrived at the main entrance just after 10:00 am.
From the highway stop, it's a few minute walk to the administration building (restaurant, tours, restroom, etc.). I canvassed the various tours, but all of the vehicular tours of the park are essentially the same and make the same four stops (petroglyphs, a shaded area with a variety of plants, and two different areas known for sandstone formations). The only decision was whether to take a minibus, overland truck or full-size bus. I opted for the overland truck since passengers are allowed to ride in seats bolted to its roof, allowing a 360° view, but all of the vehicles follow the same route, so it hardly matters. The tours follow a dry riverbed through a vertiginous red sandstone canyon searching for 1500-year old petroglyphs (several), wildlife (not much) and unusual stone formations (plenty), some of which are enormous.
The tours last about three hours. I took the noon tour, which ended at 3:00 pm, so I would have plenty of time to eat and walk to the highway to flag down the bus as it returned from Villa Unión and passed the park's entrance just after 4:00 pm.