Wow what a long time to fight that! sorry buddy... It sounds like the parallels are timing out or interfering with the network stack.
1. Are either the client or the server running on a VM or parallels or are they on the Mac side?
1a. If you are porting network traffic through a virtual network (from the Mac side into the windows side) it's likely the virtual driver is rejecting and/or timing out network packet turnarounds... Therefore it works but eventually disconnects
1b. If your porting from Mac FileMaker server to a Mac FileMaker Pro, it should in theory work, however it is not recommended to run server/client based applications (especially one as robust as FileMaker) on anything less than dedicated machines.
I use several solutions in office enviroments where the server is Mac and the clients are windows (just straight windows machines without parallels) and they are super solid and dependable. In fact filemaker dependability is hardly matched in the small to medium DB world.
That said, in the real world we have to multitask and if you have been working on this for 6 months and have an office experiencing down time as a result, it may be worthwhile to get a packet tracer (like wire shark) and watch the packets on the client and server machines from the time of startup when it work till it stops working. When you look at the network stack and how the packets route through it via a packet tracer, it will show you if its timeouts, dirty network traffic, etc.
Also note that network troubleshooting methodology says : if its working for a specific time and then stops, this is usually a result of some BER (bit error rate) threshold being exceeded in either the application, or proxy/routing architecture flow. Whatever the case, you'll find the root cause is the packets aren't able to reliably reach their targets... In this case windows and FileMaker as you stated above. I have some troubleshooting guides at my website but the answers are all at filemaker.com in the 'troubleshooting methodology' white papers too. Concentrate on your network Tom... That's where the root cause is... Otherwise it wouldnt work for anybody else right?
Lastly I will also say that we have several customers that use VM and paralells without fail, but experience has taught me to get any network applications close to the hardware NIC in the network stack as possible... The more hops a packet has to go through the more time and noise they are subjected to. Hope this helps! Have a great day Tom.