No, unfortunately it's not possible as NPM appends arguments.
You are executing: npm run method1 & npm run method2 "someArgument"
So, I guess your output is as follows:
[nodejs@mean test]$ npm run runBoth someArgument
> stackOverflow@0.0.1 runBoth /data/nodejs/node_projects/test
> npm run method1 $1 & npm run method2  "someArgument"
> stackOverflow@0.0.1 method1 /data/nodejs/node_projects/test
> ./somScript
[somScript] argument:
> stackOverflow@0.0.1 method2 /data/nodejs/node_projects/test
> ./script2.sh "someArgument"
now executing script2.sh using arg: someArgument
[nodejs@mean test]$
Also please note that arguments should be passed using "--" as follows: 
[nodejs@mean test]$ npm run runBoth -- someArgument
> stackOverflow@0.0.1 runBoth /data/nodejs/node_projects/test
> npm run method1 $1 & npm run method2  "someArgument"
> stackOverflow@0.0.1 method1 /data/nodejs/node_projects/test
> ./somScript
[somScript] argument:
> stackOverflow@0.0.1 method2 /data/nodejs/node_projects/test
> ./script2.sh "someArgument"
now executing script2.sh using arg: someArgument
[nodejs@mean test]$
But in your case, the expected output will be the same.
My package.json:
  "scripts" : {
    "method1": "./somScript",
    "method2": "./script2.sh",
    "runBoth": "npm run method1 $1 & npm run method2 "
  }
You should try another implementation (as exporting/using environment variables, or others)
Here's a possible workaround:
package.json:
  "scripts" : {
    "method1": "./somScript",
    "method2": "./script2.sh",
    "runBoth": "npm run method1 $1 & npm run method2 ",
    "runBoth2": "./alternative.sh"
  }
Where runBoth2 executes "alternative.sh" as: 
[nodejs@mean test]$ cat alternative.sh
npm run method1 $1 & npm run method2
And the expected output: 
[nodejs@mean test]$ npm run runBoth2 -- someArgument
> stackOverflow@0.0.1 runBoth2 /data/nodejs/node_projects/test
> ./alternative.sh "someArgument"
> stackOverflow@0.0.1 method1 /data/nodejs/node_projects/test
> ./somScript "someArgument"
[somScript] argument: someArgument
> stackOverflow@0.0.1 method2 /data/nodejs/node_projects/test
> ./script2.sh
now executing script2.sh using arg:
[nodejs@mean test]$
Regards