Stropharia rugosoannulata first impressed me by the size. Stamets books often show this mushroom with little kids sitting on them. They are happy in woodchip, but the most promising reward is there partnership with vegetables. Collected between Sydney and the Sunshine Coast at least. Botanical Gardens, Parks with woodchip.
Annulata refers to the ring, with repeating heart pattern
Pin heads
Cap gelatinous when wet
Gills purple, attached to stem.
Stem striated
Purple spore print
Bulbous base
Stropharia rugosoannulata growing on woodchip in a park at Samford Brisbane. June-July
There are several variations- white capped or reddy-brown capped or in the video below elaborately patterned.
mushroaming 
You seem to have done your homework. Great photographs. I am very surprised to hear of these growing in a park like that. Nice find!
Stropharia rugosoannulata is popping up all over the place, is sold in Australia as an easy edible for the home cultivator forming a symbiosis with vegetables. Some fascinating studies by Cristiane Pischl were featured in Paul Stamets book Mycelium Running. Out of the mushrooms trialed Stropharia rugosoannulata and Hypsizygus ulmarius performed the best for enhancing plant growth.
I have tried growing some here, but the culture became contaminated and it failed. Gradually, cultures like these are becoming available in Oz. They were unavailable years ago, though they were available in NZ. There was a total ban on imports. My partner bought me one of Stamets cultures as a birthday present once, not knowing about the ban, and it got stopped by AQIS.