Travis, I will try to answer some of your questions.
Your plow (model 13) was one of the Oliver #40 series of horse plows. That series of plows were built for 50 years, maybe longer. They were built with steel or a wood beams. Also, they were built with left hand or right hand bottoms. Most were painted red with some black on the moldboards, etc. The lettering was yellow. As far as the handles go, they mount different on the wood beam plows verses the steel beam models. The way you ask your question, it sounds like you have a wood beam plow. If so, there is a bracket that goes between the handles and bolts up to the back of the wood beam.
The model "A" plow is smaller than the 40 series, but most of the above information also applys.