Multiple-Inheritance does not work in JS.
Instead of building convoluted inheritance chains, though, why not try something like Mix-Ins / Traits?
Instead of trying to figure out if WalkingSpeakingPet should inherit from WalkingPet or SpeakingPet, you can instead use Traits, and add them to a pet.
function Walker (target) {
  return {
    walk (x, y) { console.log("I'm walking!"); }
  };
}
function Flyer (target) {
  return {
    fly (x, y, z) { console.log("I'm flying!"); }
  };
}
function Swimmer (target) {
  return {
    swim (x, y, z) { console.log("I'm swimming!"); }
  };
}
function Pet (type, name) {
  return { type, name };
}
function Dog (name) {
  const dog = Pet("Dog", name);
  return Object.assign(dog, Walker(dog));
}
function Parakeet (name) {
  const parakeet = Pet("Bird", name);
  return Object.assign(parakeet, Flyer(parakeet));
}
function HarveyTheWonderHamster () {
  const harvey = Pet("Hamster", "Harvey");
  return Object.assign(harvey, Walker(harvey), Flyer(harvey), Swimmer(harvey));
}