I just changed from Windows 8 64bit to Windows 8.1 64bit on an ASRock Z77 board, which worked fine, no problems at all. That board does not have any Windows 8.1 drivers available on its download page.
First understand that most if not all drivers provided by ASRock or any/all mother board manufacture are actually provided by the company that makes the various hardware components used on the board.
AMD and Intel provide the main chipset and SATA drivers. Realtek provides the drivers for their network chips, like the one used on your board. Same thing for Realtek and VIA audio chips, Broadcom and Intel network chips, the list goes on and on.
Sometimes mobo manufactures add things to drivers they get from those companies, but they are usually just additions to the driver's installation program, not the driver itself.
Windows has built in drivers that they either get from someone else (Intel for example) or write their own drivers.
All the drivers listed for your board for Windows 8 are provided by either Intel, Realtek, or VIA. I can get 90% of my ASRock Z87 board's Windows 8 or 8.1 drivers directly from Intel's download site.
BTW, what CPU are you using and are you using 32 or 64bit Windows 8.1?
I looked up an Intel made H61 (same as your ASRock board) chipset board (DH61ZE) to see what drivers they had for Windows 8 and 8.1. You can see the entire list in this link:
Note the Select an Operating System box, you can filter the driver list by OS version. There are 21 drivers for Windows 8. There are two drivers for Windows 8.1. Both of those drivers are for hardware that is not used on your board.
So Intel only has one Windows 8.1 driver for an H61 board, and that is for an Intel network chip not used by your board. If Intel doesn't have any other 8.1 drivers for their own H61 boards, neither does ASRock, at least for the Intel hardware.
In most cases, you can use the Windows 8 drivers with 8.1, or let Windows install its own drivers for your board's hardware, if it has one that is compatible.
Your BSOD problem seems to be network related, as you know. Do you know if Windows 8 has a built in driver for your Realtek network chip? I assume you are using the Realtek LAN driver for Windows 8 from your board's download page?
I also assume you are updating to Windows 8.1 with the download from the "Windows Store" 8.1 download?
Here's a link to a Realtek Windows 8 and 8.1 driver download for your Realtek RTL8105E LAN chip:
You can also try changing your network driver to a Windows driver manually with Windows Device Manager. First open up your Realtek LAN entry in Device Manager, and click on the Driver tab. Click on the Update Driver button. You can try either the automatic search option, or "Browse my computer..." for a manual update from Windows 8. If automatic fails, click the "Browse" option, and then "Let me pick from a list..." button. If there is at least one entry for a driver, or any that are labeled as Microsoft, select it and continue by following the prompts.
Not sure why you are getting that BSOD, but trying that Realtek driver or one from Microsoft/Win 8 hopefully will help.