PICKETT/BURR, Ashwell, 19th century
You don't say what sources your father consulted but there are two which will contain information which is essential if you are to make any progress. Assuming the couple were married before they left for America they presumably married between about 1848 and 1854 so with any luck this will allow you to get the marriage certificate from HALS. (If they married in the Ashwell parish church the register will contain the same details of the marriage.) The information this contains will include the names and occupations of their fathers. In addition the 1851 census for Ashwell will hopefully include both of them - if not married perhaps living with their parents - and also giving their place of birth. (Microfilms available at you nearest LDS Family History Centre - address on familysearch.)
In fact Ashwell is one of the best documented villages in Hertfordshire and there could well be important information available to you via Ashwell Education Services so that site is clearly worth a visit.
I don't know if you looked at My Ancestors Emigrated from Hertfordshire before you posted your query - but once you have collected further on William Pickett and Mary Burr fell free to let me know and I may be able to help further.
If you can add to the information given above tell me.