Welcome to Rootschat :-)
Are you sure that the marriage for Robert Elliott and Jane Robson is the right one? I am not disputing it but it would be helpful to know that you are sure they are the correct people. Perhaps you have a document for their son which gives both his parents' names. Do you have his marriage cert from when he married Margaret Sutton? That would give you his father's name, occupation at that time and perhaps family members were witnesses.
If you aren't sure, let us know what information you have that led you to this particular marriage.
If it is the right marriage for his parents, it was by Licence. Licence obtained on 1st October 1806 and the marriage took place at Chester le Street on 4th October 1806
https://www.familysearch.org/search/collection/list/?page=1&countryId=1986340scroll down that list to the items for Diocese of Durham, the Calendar of Marriage Bonds and Allegations has a subsection when you open it for Marriage Abstracts which is where I got the info from.
next item down is the Marriage Bonds and Allegations, you will find the images (probably 2) for the bond and allegation in there.
Free Reg
https://www.freereg.org.uk has a transcription of the marriage entry. A search for Elliott (and soundex in case its a single 't' in the entry) baptisms in Durham:Chester le Street brings up a list which includes the names you mention. Unlike Family Search, Free Reg transcribes all of the info in a register so you get any additional info available such as father's occupation, where they lived etc.
If you find names and dates that fit in, then go back to the list on Family Search and you should be able to find the Bishops Transcript register images - though its not searchable and you have to firkle through till you get into the right date range and then find the entry.
Hope that helps a little. Try the above and come back if you have other questions after seeing what's in the databases above.
Boo