Your only choice here is to either run a 32-bit OS or a 32-bit VM on a 64-bit host (via VirtualBox, VMware, Hyper-v, etc.).
You can, however, have more than 4GB of RAM on 32-bit Windows IF you run a server version of the OS (i.e. Windows Server 2012, 2008 R2, or 2008). These OSes allow you to use up to 128GB of RAM.
Microsoft has not made PAE available on their 32-bit desktop OSes (Windows 7/8/10), so you will only see 4GB of RAM with those no matter how much is actually installed.
It should be noted, however, that even though PAE allows you to use more than 4GB of RAM, individual applications will still be limited to 4GB each, and you still cannot run 64-bit code. You won't even be able to run 64-bit VMs, but you will be able to use your 16-bit program.