Parasoul's inverted cross on her (and Double's) necklace is a token of her religion, Trinitism. The Trinity, is a trio of "goddesses"
* (Venus, Aeon, and their Mother). The grand cathedral has stained glass window depictions, and a massive statue of the Trinity.
Parasoul does wear the Black Egret armband in many of her palettes (4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 21, 22). Of course, the umbrella symbol represents the Canopy Kingdom.
She wears the cross because, I assume, she is a devout follower of the Trinity. In Parasoul's story mode, Double relates how Parasoul's mother, the Queen, Nancy Renoir, was also quite dedicated to the religion.
Double: My, my... such anger. Your mother used to pray for peace here every day. How disappointed she would be!
* Whether the Trinity are actually goddesses is another story. They may have been like Eliza, powerful pairs of humans and parasites.