Hi,
Have you tried Google Maps, using search terms such as "Outdoor Gym", "Street workout" or "Fitness Park" (or similar)?
Try in both English and the French translations.
Far from perfect, as it's also bringing up indoor/"normal" gyms and sports stadia, but a very quick search appears to bring up a few potentially useful results.
You can’t get this equipment outdoors in my area of uk (Council budget cuts = no public toilets), fitness trail equipment has been taken away (to save maintenance costs), so it’s not just a problem when visiting France.
Day membership of a private gym?
When I could run (before serious knee & foot injuries), I’d carry hand weights. I also do ‘body control’ Pilates, which uses my own body weight but it can’t do heart-lung stuff.
But I don’t know what event you’re training for. Whatever it is, I hope you’re able to compete to an high level after your French trip.
There are usually 'trim trails' or ' fitness trails' in many parks / leisure areas (base de loisirs) in most towns in France, however I don't know whether they would be what you are looking for.
Maybe contact tourist offices of the places you intend to visit, to ask them the question directly.
The Office du tourisme is a good shout, though I do wonder how they keep track in such a disparate category. Either way, good to know French municipalities still invest in outdoor fitness infrastructure
Google Maps and Mappy feel much thinner than Calisthenics Parks and IronSpot when it comes to street workout spots and outdoor pull-up bars. However,, those last two have good spots near where I'm headed, so I'll work with those.
Amazing family weekend with old steam engines, classic car displays, market stalls, and full catering and bar. And camping on site - Save £25 by booking in advance.