Until Pokemon Home compatibility, you can't with traditional breeding methods.
But now with the Camp mechanics, there's another way to transfer Egg Moves between Pokemon.
Step 1 is to get the Pokemon you want to learn the Egg Move, and make sure it has an empty spot in its 4 move slots. Then give it the new Mirror Herb item to hold, which can be purchased at Delibird Presents.
Step 2, get your target Pokemon with the empty slot and the Mirror Herb (in this example, Shroodle or Grafaiai), and any Pokemon that already knows the Move you want to transfer (in this case, Parting Shot). With both of these Pokemon in your party, set up a Camp. Now pack up the Camp. The target Pokemon (again, Shroodle or Grafaiai in our example) should have immediately learned the Move in question (again, Parting Shot). Take back the Mirror Herb from your target Pokemon.
The best part is that once you have the Egg Move on the target Pokemon, that Pokemon can now breed down the Egg Move as normal. Both methods function side-by-side. I'm not a fan of how eggs work, personally, but Egg Moves got dramatically simpler with the Camp.
