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Because that's how the morphing works. You get the pieces to make the pet, then you morph it into the mount. You have to recollect the pieces for the pet again now that you'll be able to morph into, looks like, an outfit next.
The pet is used in the "construction" of the mount. You'll need to gather parts for another one, and that one will too be consumed when you unlock the next reward (an outfit IIRC).
Since Blackwood this has been the "formula":
Q1 - buy fragments for pet, create pet
Q2 to Q4 - buy fragments for pet and award, once you have all the award fragments combine them with the pet you already have and create new pet if you have enough fragments for it. Repeat until last quarter.