Hi ...
Welcome to Rootschat!
Yarnton parish records date back to 1569, so there's a chance that you may find information on Thomas snr. However, I have checked quite a few Oxfordshire resources and there's very little on DUNSBYs prior to 1700.
I did come across this marriage on the OFHS's Oxfordshire Marriage Index (1538-1837) which may be worth investigating further:
15 JUN 1718 - Thomas DANSBY (of Farthinghoe) to Sarah SAUNDERS (of Yarnton) at Oxford St. Mary Magdalen(The OMI includes the parish of Yarnton, so it would seem that there are no pre-1700 marriages there.)
Farthinghoe is in Northamptonshire and about 20 miles north of Yarnton, and it's always possible that Thomas jnr.'s parents married after they had their children! Additionally, Yarnton is right on the old (pre-1974) county boundary with Berkshire. The OFHS have a separate marriage index for those parishes previously in Berkshire (but now in Oxfordshire) and you can request a look-up via their website at
www.ofhs.org.uk (there is a small fee).
You may also be interested in this article on the Yarnton village website at www. yarnton-village.org.uk/grandmother-tale/ - go the paragraph starting "Growing in the lane ...". This would seem to indicate that the DUNSBYs originated from Lincolnshire - the IGI has a good number of pre-1700 DUNSBYs listed for that county.
Hope the above gives you some more leads to follow!
Pennie