I think a download manager is your only option. Firefox can do more but limited to 15 from my Googling.

From my testing just now it seems 6 is fine and the 7th sits there queued and will start when one of the first 6 finishes. There was a bug report that this should be better reported I found just now.

Really you should not be hitting a web server with more than 2 connections at once to be compliant but most browsers ignore that....