In addition to what Sandy has already said, and to clarify how you need to approach this yourself and what you need to purchase, I would make the following points:
1. Obtain a copy of Eliza Smith's death certificate (Sandy has confirmed the details) from the site that Orpheus has provided you a link to.
2. Obtain marriage and burial searches from Bucks FHS for the same time period you obtained baptisms.
3. Check that Thomas Smith, son of Thomas and Maria, baptised in Thornton in April 1822 did not die as a child.
Just because Thomas Smith, with wife Maria, is listed as a labourer on his children's baptisms does not mean he is not the correct Thomas Smith. On the 1841 Census, one year before your Thomas married Eliza Read, he was listed as a gamekeeper. Somebody working as a labourer could quite easily become a gamekeeper and then revert back to a labourer again.
Assuming Thomas Smith baptised Thornton in April 1822 is not buried in Thornton before 1842, then this will most likely be your Thomas for the aforementioned reasons. The 1851 Census entry in Passenham with second wife Hannah lists him as a cordwainer born c1821 so all the details agree with his 1842 marriage. As Marilyn has given you the details of his second marriage, there is no need to obtain a copy of the certificate and I did not say you needed to. At no point does this Thomas Smith, shoemaker, claim to be born in 1817 - you are confusing him with the Lillingstone Dayrell Thomas who is NOT yours.
4. You need to confirm Thomas Smith's mother Maria's maiden name. The Daniel Adams with them on the 1841 Census may be her father but unless the baptism of Thomas Smith in 1822 lists his mother's maiden name (not impossible but unlikely) you will need to purchase the birth certificate of her youngest child, Joseph, to confirm her maiden name. This would appear to be the correct details:
Joseph Smith Sept qtr 1840 Buckingham (Registration District) Vol 6 Page 316
You can order this from the previously mentioned GRO website.
Smith is too common a surname to take a guess at the correct marriage without further checking.
5. When you have the birth certificate, you can then check the marriage search results to confirm where and when Thomas and Maria married. This will not tell you either Thomas or Maria's father's name because it will have occurred before 1837 and the start of civil registration.
6. Thomas senior, gamekeeper and labourer, remarried before 1851. As Sandy has mentioned, you need to obtain a copy of this marriage certificate to conclusively confirm his father's name. The details appear to be the following:
Thomas Smith March qtr 1850 Winslow RD Vol 6 Page 485
Tattenhoe, where Thomas senior is living with his two youngest sons in 1851, is in Winslow RD. Although it would seem he intended to marry Sarah Mitchell in 1847, it does not look like they actually married until 1850. This seems the most likely marriage certificate but you would need to consider the details on it with other information from the Census, such as occupation and residency.
7. The Thomas Smith baptism you mention in Whaddon in 1796 is more than likely Thomas the gamekeeper but the marriage certificate should hopefully confirm this. You can then look for his parent's marriage from the search results from Bucks FHS.
10. When you have Maria's surname from her son Joseph's birth certificate, you can then obtain baptism/marriage/burial searches from Bucks FHS for that surname to look for her parents. She is listed as born in county on the 1841 Census so hopefully you should be able to find this.
As Sandy has said, we are here to help you with your research but you need to obtain what has been suggested.
Nicola
