Sorry for late answer, i rarely come here
yes, you need a valid subscription... my first language is not english, so it's a little complicated for me to explain the steps but if you use Chrome, you just need to use his developer tools and to look for all the packets that are transmitted... one of them will be the subtitles that you need. Then you just need to import them in another tool like Subtitle Edit to convert them in SRT.
The link that you provided, in Netflix Italy, is for an episode of How I Met Your Mother and the only subs that i could extract, would be the italian ones... (i should use a VPN to get an american IP to connect to Netflix USA to see also the english ones but i am little lazy to do it...)