Thurid's triggered ability doesn't trigger if a Pegasus, Unicorn, or Horse creature is put onto the battlefield without being cast.

As a copy of a permanent spell resolves, it's put onto the battlefield as a token rather than putting a copy of the spell onto the battlefield. The rules that apply to a permanent spell becoming a permanent apply to a copy of a spell becoming a token.

The token that a resolving copy of a permanent spell becomes isn't "created." Abilities that refer to a token being created won't interact with the copy resolving.

Thurid's triggered ability and the copy it creates will resolve before the spell that caused it to trigger. They resolve even if that spell is countered or otherwise leaves the stack without resolving.

Thurid's triggered ability will copy only creature spells, not noncreature kindred spells, even if those spells have the Pegasus, Unicorn, or Horse types.