Doubling Season and Yarok, the Desecrated have no direct interaction. Doubling Season's ability creates a replacement effect. It is not a triggered ability on which Yarok's ability could trigger.
As per this Q&A, Doubling Season affects permanents as they enter the battlefield with counters already placed. These static abilites create replacement effects, which are different from triggered abilities. Yarok only affects affects triggered abilities.
603.1. Triggered abilities have a trigger condition and an effect. They are written as “[When/Whenever/At] [trigger condition or event], [effect]. [Instructions (if any).]”
Replacement effects have a variety of wordings. The most common wordings that apply to permanents are:
614.1c Effects that read “[This permanent] enters the battlefield with . . . ,” “As [this permanent] enters the battlefield . . . ,” or “[This permanent] enters the battlefield as . . . “ are replacement effects.
614.1d Continuous effects that read “[This permanent] enters the battlefield . . .” or “[Objects] enter the battlefield . . .” are replacement effects.
614.1e Effects that read “As [this permanent] is turned face up . . . ,” are replacement effects.
These are affected by Doubling Season if they involve placing counters on a permanent, but do not trigger Yarok.
If counters are placed on an object due to a triggered ETB ability, Yarok will double that ability, and Doubling Season will double both effects, for example if you choose the +1/+1 counters on Knight of Autumn
Doubling Season does not directly interact with Yarok, becasue it creates a replacement effect, but does not trigger at anything. They do each apply independently of each other.
Example 1: Barkhide Troll is successfully cast. Doubling Season sees the creature trying to enter with one or more counters, so it doubles the counters as it enters. Yarok sees no triggered abilities and does not interact. Barkhide Troll enters with 2 counters.
Example 2: A Planeswalker is successfully cast. Doubling Season sees the Planeswalker trying to enter with one or more counters, so it doubles the counters as it enters. Yarok sees no triggered abilities and does not interact. The Planeswalker enters with twice its starting loyalty.
Example 3: Hada Freeblade is successfully cast. As it enters, it triggers it's own ability. Yarrok sees an ETB ability, and makes it trigger an additional time. As each of the triggers resolve, Doubling Seasons sees that one or more counters are being placed on a creature, and doubles the number of counters being placed on the creature. Hada Freeblade has 4 counters after all triggers resolve (2 triggers * 2 counters per trigger = 4 total counters).