Most VPN software isn't captive. Meaning if you VPN to a remote network B 192.168.2.0/24 then it will add a route only for that specific remote subnet. Likely it will also add a virtual IP for that network. So if you do a route print from the command prompt, you'll see a route similar to:
Network Destination: 192.168.2.0 Netmask: 255.255.255.0 Gateway: 192.168.2.25 Interface 192.168.2.25 Where .25 is your VPN virtual IP.
What this means is that any traffic destined for 192.168.2.1-255 will go through the VPN and everything else (including destination 192.168.1.X) will just go to your default gateway and not reach either remote site. Then your ISP will just drop any packets that have a 192.168.1.X destination.
What you want is for both subnets to route through the VPN. Usually you can change this in the VPN client software (remote networks is the common term). Alternatively you can add a route that will tell your computer to use the VPN for both remote networks.
route add 192.168.1.0 mask 255.255.255.0 192.168.2.25
So now all traffic destined for 192.168.1.1-255 will be sent through the VPN rather than out to the internet. So it should hit your router at site B and then be sent through your site-to-site VPN to site A.