Examples:
Happy
Sad
Comedic
Dramatic
Cartoonish
Or absolutely anything else that makes your pet uniquely them!
These are very simple examples but you can build on this or try a blended style - just please bear in mind that if you blend tones or styles that don't make sense or are contradictory, the song may sound just as nonsensical or unhinged. i.e. blending something cartoonish with something ethereal.
You're welcome to experiment, but if your franken-song becomes so confusing that your parrot unlearns how to talk, any replacement song would require a separate purchase.
My suggestion (for what it is worth) is maybe just pick one tone so that the outcome is more stable.