MrsHal--
Glad to see you already made it to the "Researching..." forum. I understand from your post that you have info on your Civil War ancestor but are looking for further info on his line; i.e. his parents. Most Civil War records don't list parent's names, but some do. It depends on the State, the type of record, etc.
To start on any possible Civil War ancestor, I normally use the Nationa Park Service's Civil War Soldier and Sailor Database at
http://www.nps.gov/civilwar/search-soldiers.htm I tried that site for your ancestor but didn't come up with any listing. Unfortunately, that site is very dependent on spelling, so you might have to try multiple variations of spellings of the last name to find the one used in the records.
Do you have access to Ancestry.com? That can normally help track the family line. I did a quick search on Ancestry and found that a W. R. Brusselman (aka William R. Braselman) was a Private in Company B, 5th Louisiana Infantry. An 1860 LA census shows Wm R Braselman as being born in South Carolina in about 1822 (sounds good); he was a cotton classer and had a wife Mary F (age 33 from LA) and two daughters, Agnes A R (11) and Myra M(athilda) J (9). I assume this is your ancestor, correct? Another ancestry tree doesn't have William's parents but lists his wife's maiden name as Mary Fanelie Robertson (~1830 - 25Dec1890). They married 31 May 1848 in LA. William died 23 Oct 1877.
A William R. Braselman was also Capt. in 6th Regmt, PF Smith's Brigade of LA Volunteers 1846-48 during the Mexican -American War.
Unfortunately, I don't see anything in Ancestry that quickly points me to his parents. One thing you can try (no guarantee) is to order his compiled service records from the National Archives. However, it's not highly probable that those records would contain info on his parents.
If you have any questions regarding William's Civil War service, we can surely help give you some answers there. Keep trying and be patient. We might find someone on this forum who has the key to your puzzle.