What exactly is a "parenthetical" reference??
Google, the all-knowing overmind, will tell you. It's just a method of citing your sources. Some citation styles use footnotes or endnotes, but most use parenthetical references, or at least give you the option of using them. Unfortunately, this means you will need to check with your teacher to see which style you are supposed to be using -- MLA, APA, Chicago, Harvard, etc. Then, once you know what style you need to use, Google it to find out the correct format.
And section headings for the research paper would be like one of those five topics of capital punishment. So for example, as a heading I would put History of the Death Penalty??
That would work, and make it easy for the reader to follow your essay.
It seems as though a thesis statement is stating what you believe is right and I know your really not suppose to do that in a research paper except in the conclusion maybe. So how do I go about writing the thesis statement in a research paper?
Well, usually you would still argue something; you would just argue it with only reference to research sources, instead of injecting your own personal opinions in to the essay. Check out: esc.edu/htmlpages/writerold/menud.htm for more details. If you absolutely aren't allowed to do that, though, you could always craft a research question. So, start out with a question that you hope to answer through your research instead of a thesis. Then, in your conclusion, you simply have to write down your answer based upon the research you did. So, think of what your thesis would be if you were allowed to have one, then turn it into a question in the intro, essentially.