Chain Drive's page is here.
No Doves Fly Here lasts from 10PM - 2AM, every first and third Wednesday of the month, starting October 19, 2011. As of now there is no door or cover. You must be over 21 to attend.
The following is from Chain Drive's page, regarding parking:
[Chain Drive] is one block south of the Cesar Chavez/Red River St. intersection in downtown Austin.
Free parking is available in our own parking lot and on certain areas on Willow St.
If you get to the neighborhood early, there is parking in many on-street spots on Driskill and Rainey Streets also. Please respect our neighbors and do not park in their lots or block their driveways.
LAZ Parking has terminated their agreement with Chain Drive allowing our customers to park for free in the lot next to IHOP. They now charge $5 but do offer a monitored facility three nights a week that offers a greater degree of protection than before.