Hi there,
I don't wish to dampen your enthusiasm in any way, but I thought that as you appeared so interested that you should know something about what you are asking. The surname of Parsons has English roots and, as such, there is no such thing as "the Parsons family coat of Arms". Arms were/are granted to a person, not a family. To be able to claim any right to display those arms as your own you would have to provide documented paternal ancestral research for each generation back to the originally recorded armiger (person bearing arms). Each generation would also have to be researched for all the children of a marriage to determine if the ancestral link was through a cadet branch of the family or the main bloodline.
As Stephen has mentioned, there are 13 entries for the surname of Parsons in Burke's General Armory and another two entries in General Armory Two. Just which ones are you really interested in? As Stephen mentioned, the arms that you mentioned became extinct in 1764 - which means that the male line of armigers for this coat of arms died out in 1764, so you couldn't have any claim to be able to use them. To be able to understand what I am getting at, may I direct you to read the very informative phamphlet produced by the Society of Genealogists in the UK which can be found at <<
http://www.sog.org.uk/leaflets/arms.html >>.
Surnames are not part of an achievement of Arms - the only writing to be found outside the shield/crest area that is part of an armorial achievement (Arms) is the Motto and, in some cases, a 'crie de guerre' (war cry).
Regards,