I've done a bit of searching but I can't find anything conclusive.
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The equipment in your picture may be coated with a couple coats of an oil-based, penetrating stain. Or it may be just bare wood. Cedar and redwood, for example, stand up to weather very well on their own.
Much other playground equipment is made from regular pressure-treated lumber.
Before the early 2000s pressure treating lumber involved arsenic, but since then arsenic has been replaced with copper compounds which are said to be completely safe according to government and playground manufacturer industry groups, etc.
Much construction of playground equipment uses pressure-treated softwood (wood from conifer trees).
Current treatment of choice is ACQ (alkaline copper quaternary) but there is also CA (copper azole), SBX (sodium borate), and MCQ (micronized copper quaternary). In America the species most commonly treated is southern yellow pine. In other parts of the world it is usually other species of pine or fir, in Canada western hemlock is also common.
First clue is that it is 2x or 'two by' material, 1x6, 2x4, 4x4 etc. same as you get from any lumber source. Other than the usual profiles usually you can easily identify such wood by eye because softwood has obvious light/dark grain patterns. And often many knots may be present! This wood may have an artificial greenish or brownish colouring from pressure treatment.
Some elements in the playground photo are made from such material, visible above the round table on the left of the image. Also the blackboard frame.
Other elements in this photo are not pressure-treated wood. The sawn log on the ground 100% because bark is never left on during treatment, but the standing logs are also very unlikely because I think they never treat natural-shaped logs.
The logs possibly are species naturally resistant to decay outdoors, there are a few such species in the US, including white oak, black locust and sassafras.
For dry and hot climates decay-resistant species may not be necessary to give long life because water is needed for decay.