With the latest update for Android, WhatsApp has come up with a totally new UI and improved material design. After I updated to version 2.12.84, I could not find the option to save profile pictures of my contacts to Gallery. But the old method still works, after viewing full size pic, I can drill down to my file manager and can grab the pic from there.
Downloading DP on Desktop
With help of web.whatsapp.com you can download high quality profile photos on your desktop. Head on to the site and scan the QR code shown there with WhatsApp on your Android/Windows phone. This feature is not yet enabled for iPhone users.
Once you have done loading the page, Open a new tab with this link
Use the above link and replace the phone number with your friend's number. It should include the country code without any "+" sign. Like here, we have 91 as country code followed by the 10 digit mobile number.
The other parameter, you can see is t=l. It represents the size. Passing t=l will load high resolution image (640x640). If we change it to t=s, it will load small size image (96x96).
The method will work only when you have opened the WhatsApp Web on the browser and session is connected.