I'm guessing that you have the Sony amp hooked up to the 'MAIN OUT' of the receiver. This WILL NOT WORK! That main out connection is volume controlled. Most of the time, you would get a two-zone AV receiver which would separate the audio completely for the family room vs. a secondary room. It looks like your receiver is a TWO ZONE unit. It has a set of RCA outputs on it that are labeled 'ZONE 2'. Those RCA connections have completely separate volume control from your 'MAIN OUT' connection which is tied to your main zone.
If you want the audio on all the time and tied to the main zone, then move the connection that goes from the Yamaha to the Sony to your VCR2->OUT connection. This output is full voltage and will send full line level audio to the Sony amplifier at 100% volume. From there, you should turn the volume down in the individual zones to an appropriate listening level. But, the way that receiver is designed, the ZONE 2->OUT RCA connections are a better way to go.
Your issue, if I'm guessing about your setup, is entirely in the fact that you're feeding the amplifier from the 'OUTPUT MAIN' RCA connections on the back of your Yamaha, and that means you are not feeding enough signal to the Sony for your speakers at all times.