I was born in Germany to non-German parents. However, when I was 6 years old my father acquired German citizenship. Does that mean that I can automatically receive a German passport because my father has one?

That depends how long your parents lived in Germany before you were born.
Contact the German embassy and ask them if you think you qualify.
http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/__Citizenship.html

http://www.bmi.bund.de/SharedDocs/Standardartikel/EN/Themen/Migration/Staatsang/faq.html;jsessionid=566CD23D1FD21FC8C29955C1AF9694 DB.2_cid295?nn=267412#doc1308132bodyText2
Section 4, para. 3 of the Nationality Act

In addition to the principle of descent, since 1 January 2000 German nationality law also recognizes the principle of birthplace (in Latin: soli) for the acquisition of citizenship. According to this principle, children born in Germany to non-German parents may, under certain conditions, acquire German citizenship. These conditions are as follows: One parent has to have lived legally in Germany for at least eight years prior to the child's birth and be neither an EU citizen entitled to freedom of movement, an equally entitled citizen of an EEC member state, a Swiss citizen entitled to freedom of movement, or the holder of an EU residence permit or a settlement permit in accordance with the ResidenceAct.

I was born before the year 2000, so the new law does not apply to me, unfortunately. My Dad had lived there for 15 years prior to my birth and my mother lived there for 6 years prior to my birth. I am aware that children born after the year 2000 in Germany can get German citizenship regardless of their parent's origins, but I have not found any laws regarding citizenship for children whose parent(s) received German citizenship after their birth. I am concerned because I am currently living in America and would like to move back to Germany.

That depends how long your parents lived in Germany before you were born.
Contact the German embassy and ask them if you think you qualify.
http://www.germany.info/Vertretung/usa/en/05__Legal/02__Directory__Services/02__Citizenship/__Citizenship.html

http://www.bmi.bund.de/SharedDocs/Standardartikel/EN/Themen/Migration/Staatsang/faq.html;jsessionid=566CD23D1FD21FC8C29955C1AF9694 DB.2_cid295?nn=267412#doc1308132bodyText2
Section 4, para. 3 of the Nationality Act

In addition to the principle of descent, since 1 January 2000 German nationality law also recognizes the principle of birthplace (in Latin: jus soli) for the acquisition of citizenship. According to this principle, children born in Germany to non-German parents may, under certain conditions, acquire German citizenship. These conditions are as follows: One parent has to have lived legally in Germany for at least eight years prior to the child’s birth and be neither an EU citizen entitled to freedom of movement, an equally entitled citizen of an EEC member state, a Swiss citizen entitled to freedom of movement, or the holder of an EU residence permit or a settlement permit in accordance with the ResidenceAct.

I was born before the year 2000, so the new law does not apply to me, unfortunately. My Dad had lived there for 15 years prior to my birth and my mother lived there for 6 years prior to my birth. I am aware that children born after the year 2000 in Germany can get German citizenship regardless of their parent's origins, but I have not found any laws regarding citizenship for children whose parent(s) received German citizenship after their birth. I am concerned because I am currently living in America and would like to move back to Germany.

Just contact the embassy and ask. They are the only ones that can give you a definite answer.

Thank you for your response. I made an appointment with the embassy.