Hi Graham
I've looked through the records for you and unfortunately haven't been able to find any of the info you requested. There aren't a lot of either Allards/Allwoods or Potters in the records but I'll detail the ones I did find.
I checked Baptisms and Burials for the period 1730 - 1799 and Marriages for 1730 - 1774. I should point out that quite a lot of the entries up to about 1740 are nearly unreadable - only thing to do is look at the [very dim] scrawl and see if it looks anything like the name you're looking for!
Okay, Potters first as there were more of them. I don't think the John and Susan Potter will be your Mary's parents as they have a daughter called Mary in 1772 BUT John could maybe be her uncle?
1732 Alice, daughter of Philip and Ann Potter, bapt. Dec 3
1734 Priscilla, daughter of Philip and Ann Potter, bapt. Dec 11
1735 Priscilla, daughter of Philip and Ann Potter, buried in woolen Aug 31
1735 Alice, daughter of Philip and Ann Potter, buried in woolen Sept 7
1738 Richard, son of Philip and Ann Potter, bapt. Feb 22
1754 John Potter and Susan Martin, both single of this parish, married entry no. 2 Nov 17
1755 Susan, daughter of John and Susan Potter, bapt. Nov 23
1757 John, son of John and Susan Potter, bapt. Oct 2
1759 James, son of John and Susan Potter, bapt. Feb 5 or 8
1764 Thomas, son of John and Susan Potter, bapt. Apr 18
1765 Elizabeth, daughter of John and Susan Potter, bapt. July 21
1765 John, son of John and Dusan Potter, buried Sept 5
1767 Elizabeth Potter, an infant, buried Aug 2
1767 Elizabeth, daughter of John and Susan Potter, bapt. Dec 24
1769 Ann, wife of Philip Potter, buried Aug 4
1772 Mary, daughter of John and Susan Potter, bapt. Jun 14
1773 James Allwood and Mary Potter, both single of this parish, married entry no. 129 May 17
1775 Philip Potter, buried Jan 11
1777 Richard Potter, buried May 21
1788 John Potter, pauper, buried Jan 10
ALLWOODS
1773 John and James, twin sons of James and Mary Allwood, bapt. Dec 2nd
1773 John Allwood, buried Dec 17 (NB: doesn't say 'infant')
1775 Elizabeth, daughter of James and Mary Allwood, bapt. Apr 23
1777 John, son of James and Mary Allwood, bapt. Mar 23
1778 William, son of James and Mary Allwood, bapt. Sept 27
1780 Sarah, daughter of James and Mary Allwood, bapt. Nov 26
1782 Joseph, son of James and Mary Allwood, bapt. Dec 22
1786 Mary, daughter of James and Mary Allwood, pauper, bapt. Feb 12
1788 Edmund, son of James and Mary Allwood, pauper, bapt. Nov 2
1790 Jeremiah, son of James and Mary Allwood, pauper, bapt. Sept 3,5 or 8
1793 Henry, son of James and Mary Allwood, bapt. Feb 2
1799 Elizabeth, daughter of John and Susan Allwood, bapt. June 23
I assume you already have James and Mary's children but I've listed them anyway so you can check with what you may have.
Do a google search for the Suffolk Family History Society. Before I bought the records someone did a free search for me so I knew I was in the right area. It could be that your family were between 2 churches and someone at the site could give you a likely church (Brockford?) to check. Here's one link
http://www.genealogy-links.co.uk/html/sfk.lookup.htmlSorry I couldn't find what you were actually looking for but if any of this ties up with your family and you need something else checking just ask
Cheers, Debbie