This is really weird ?!??
I was so surprised by your last post that i've decided to check by myself. I didn't manage to reproduce your parse error, but i managed to make BuddyDrive works on a multisite config with BuddyPress activated on a blog different than blog_id #1
To be precise, it's blog_id #2 ;)
So i've created a new install of WordPress multisite, went to wp_config.php and added :
define( 'BP_ROOT_BLOG', 2);
I also set WP_DEBUG to true
Then, i went in blog #2 Dashboard and activated BuddyPress 1.7. I checked on the front with the default theme activated (TwentyTwelve) and BuddyPress is up and running.
Then i simply deleted line 10 of BuddyDrive's main file (/wp-content/plugins/buddydrive/buddydrive.php) which means the network tag is not there anymore.
Then i went in blog #2 Dashboard and activated BuddyDrive, my beautiful welcome screen displayed as it should, i love the welcome screen of plugins ;)
Then, i went on front on the user's nav 'BuddyDrive', and added a file. Everything went just fine !!! No errors, the file uploaded in /uploads/site/2/buddydrive. Then, i've checked my WP embed functionnality, and again it worked. Finally i tried to download the file, and again success.
So i'm sorry, everything seems to work fine on your particular config. I suggest you test it by yourself doing again the different steps on a WordPress 3.5.1, BP 1.7 fresh installs using for example MAMP or WAMP or XAMPP, and you'll see it's working ;)
I'm sorry, i have no time to check for all cases of WordPress configs (for instance, i dont know if a 2.7 WPMU upgraded till 3.5.1 is working)
By the way having the error your mentionned is very weird because if you check the explorer template at line 9 we have an if statement.
I hope you didn't misunderstood my last post and deleted line 9 of the explorer template
The line to remove is in plugin's main file header > the network tag is to be deleted, i use this tag to force the plugin to be activated in network admin.