Per the title, pets of any and every sort contribute to the SR timer when they die or despawn. Depending on the classification-tier of the pet, the significance of this contribution may vary.
In the clip below, Swarmlings have been modified to have their monsterClassification be ‘Boss’ for the sake of demonstration. No other changes have been made to their pet data, nor has the spawn skill been changed (or included in the mod, for that matter).
Note, of course, that it is not merely a display issue - the timer does indeed increase upon new deaths of Swarmlings.
In my testing this issue seems to extend to all pets, vanilla or otherwise.