The HGTV couple who say their dog was killed while supposedly in the care of a professional pet-sitter is demanding accountability ... and, they're thinking about ways they can redefine the legal status of dogs in Florida.

We spoke with Eilyn and Ray Jimenez -- best known for starring in the hit reality show "Divided by Design" -- on "TMZ Live" Friday ... and, they ran through the events surrounding their dog's death, which they previously laid out on social media.

ICYMI ... Eilyn and Ray were told their dog Aria -- a 12-year-old Shih Tzu -- died of a heart attack while in the care of a pet-sitter while they were out of town for 2 days. However, they said they learned from the cremation center where their dog was taken before they got home that their pup was actually killed by anot

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