Why is it my responsibility to pay a supplemental bill? Why isn't it paid out of my escrow account?

Supplemental tax bills are not part of the real estate taxes, so they are the homeowner's responsibility.  For example, the state of California will only send the supplemental tax bill directly to the homeowner. 

If supplemental taxes become severely delinquent and are considered a threat to our interest in the property, we will pay them as a one-time disbursement from escrow. We will then include the amount we advanced in the next escrow analysis.