How can I run multiple npm scripts in parallel?

Use a package called concurrently.

npm i concurrently --save-dev

Then setup your npm run dev task as so:

"dev": "concurrently --kill-others \"npm run start-watch\" \"npm run wp-server\""

If you're using an UNIX-like environment, just use & as the separator:

"dev": "npm run start-watch & npm run wp-server"

Otherwise if you're interested on a cross-platform solution, you could use npm-run-all module:

"dev": "npm-run-all --parallel start-watch wp-server"