Adobe's 64-bit Flash Player support for Linux (and also Windows and Mac OS X) is limited to their Flash Player "Square" Beta, which is available from Adobe Labs. The latest 64-bit Square Flash Player beta from Adobe is their "Preview 3" release from last November. This release is based upon their older Flash Player 10.2 technology and not Flash Player 10.3.
The good news is that we're told by a source with Adobe Flash Player 10.4, the 64-bit "Square" support will be syncing with the latest code-base where the 32-bit support will be residing. Hopefully this means the 32-bit and 64-bit binaries will remain on par, but it wasn't certain from these communications. But with Flash Player 10.3 having been just officially released last week, it will likely be a few months before the next release emerges. The 64-bit Flash Linux news was shared earlier today on my Twitter feed.
Beyond the 64-bit syncing, it also appears the next major release of Adobe's Flash Linux Player will begin to support VA-API for video playback acceleration. With Flash Player 10.2 is where Adobe introduced NVIDIA VDPAU Flash acceleration so that Flash video content was offloaded to the graphics processor for NVIDIA drivers supporting the Video Decode and Presentation API for Unix. While VDPAU is a great interface, it's not supported by the proprietary AMD Catalyst driver, the Intel Linux driver, or the other open-source graphics drivers. Instead these drivers normally target VA-API.
With the next major Adobe Flash Player release is where we'll likely see this VA-API support land so that HD video content is offloaded to the GPU for more than just NVIDIA GPUs being controlled by their binary blob. Under Windows and Mac OS X, Adobe already supports video playback acceleration for all of the major graphics vendors.
VA-API is natively supported under Linux with the open-source Intel driver and with the proprietary AMD Catalyst driver via the VA-API to XvBA library created by Splitted Desktop Systems.