You will not find a pets allowed field in the property data the HomeRunner API returns.
Why this happens. Pets allowed is not a field. It is a taxonomy term under property_flag, and the property payload returns bare term IDs rather than names.
What to do. Fetch https://<your-site>/wp-json/wp/v2/property_flag to see the list of flags and match each ID to its name. Most properties also carry an unrelated flag, such as "standalone", so do not read anything into it.