Fees and charges are only synced with Airbnb and Vrbo upgraded connection. For other connected channels, include Vrbo standard connection, fees should be updated directly on the channel.
To update your pet fee, follow the steps here to access your listing's pricing settings. Scroll down to the "Standard fees and charges" section, and next to "Other standard fees" click Add new fee. Scroll down in the pop-up list of fees and select Pet fee. See here for more information on setting your standard fees and charges.
If a Pet Fee is added with zero amount charged, this will be displayed as "Pets stay free" in Airbnb.
In addition to adding the pet fee, you need to allow pets in your listing settings. Follow the steps here to access your listing's booking settings. Under House Rules, activate the toggle for "Pets allowed". See more information here.