Farsi wikipedia gives a number of different ages for those paintings (please note that my Farsi is not particularly good and English is just my second language, so the translation is probably imprecise in several places):
غار دُوشِه غاری با قدمت بیش از ۸ هزار سال است که ۱۱۰ نقش و نقاشی به وسیله انسان‌های اولیه بر روی آن کشیده شده‌است. این غار در اطراف روستای کورشوراب علیا شهرستان دوره چگنی در غرب استان لرستان واقع شده‌است. وجود این غار نشان می‌دهد که سابقه دیوارنگاری در ایران به دوران نوسنگی و حدود ۴۵۰۰ سال پیش از میلاد بازمی‌گردد.
Dousheh cave is a cave with a history of more than 8 thousand years in which 110 drawings and paintings were drawn by early humans. This cave is located in the vicinity of the village of Kurshurab-e alia[?] in Dure Chagani[?] county in western Lorestan province. The existence of this cave shows that early cave painting in Iran returned[?] during the neolithic and around 4500 years BC.
(FWIW Google Translate renders the last sentence as "The existence of this cave shows that the history of murals in Iran dates back to the Neolithic period and about 4500 BC." which definitely sounds better, but my online dictionaries do not really agree that you can translate بازگشتن to "date back to")
The caption below the image of the rider reads
دیوارنگاری در غار دوشه، لرستان، حدود هزاره هشتم پیش از میلاد.
Cave painting in the Dousheh cave, approximately eighth millenium BC.
These numbers are a bit hard to reconcile (although "eighth millenium BC" might just be a confusion with "8 thousand years [old]"), which makes me wonder how reliable any of them is.
Anyway the article and also the linked sources do not mention horse riding, which kind of raises the question how certain the identification of that figure with a horse rider is. Might he/she just be standing next to the horse (or whatever animal that is) and touching its back?
There actually is some academic cooperation between Iran and other countries (except Israel and probably the US?), and lots of academically educated Iranians leave their country for western Europe (and other places) every year. So while international isolation might be a factor here, I do not think it is the most important one.
(P.S. since I have mentioned Israel in connection with Iran's political isolation, it might be worth pointing out that Iran's isolation from Israel is mainly imposed by the Iranian side.)