There are lots of references to this law, but it may be directed toward the drivers or passengers of motorcycles.
Rumor has it that the penalty ranges from a slap on the wrist to a ticket for a few hundred baht. Thankfully the exchange rate is favorable and this is around $10 US.
Section 3-6 Animals killed or injured on rights-of-way. Notwithstanding any provisions herein to the contrary, animals injured or killed on or along public streets or rights-of-way shall be presumed to be or having been running at large. The animal control officer shall remove all such animals and, at his or her discretion, deliver those needing…