I also checked the images from the parish register on ancestry, definitely no Wood or Woodward born in 1620 or the years on the page before and after.
There is also no marriage between a Thomas French and a Sara in 1636. There is just one wedding recorded in that year .
The first child of Thomas and his wife Sara was Patience baptised in Nether Heyford ;16th of September 1637 (actually the entry is in latin, her name is written as Patientia ) so they may possibly have been married the year before but as to where?
edit realised I had written Sarah with an h but it was Sara in the baptismal entry I viewed