I am as yet unsure about Thomas. There were two, or possibly even three, Thomas Byrds living in the parish of Youlgreave at about the same time, and as yet I haven't established which one was the father of Francis.
One, or possibly two, of the Thomases is found in the will of Roger Bryde of Youlgreave dated 1549 (who is possibly the same Roger who is recorded in the records of the Duchy of Lancaster as being aged 46 in 1528), in which he mentions his son Thomas and his three children, one of whom is a son also called Thomas. (Roger's grandson is the main candidate here.)
The other Thomas is found in two seperate wills. The first is the will of William Bird of Stanton Hall dated 1578, where he mentions his (younger) brother Thomas. The other is the will of another William Byrdes of Stanton Hall, son of the above William, dated 1593, which mentions his Uncle Thomas Byrdes.
There are two burials in Youlgreave parish register. One is Thomas Byrd, husbandman, on 9/1/1586. The other is Thomas Bird, "constable and a fine neighbour", on 13/8/1599. Given that the Thomas who was brother and uncle to the Williams of Stanton Hall, was still alive when the younger William made his will in 1593, he must be the one buried in 1599, meaning the one buried in 1586 is probably the grandson of Roger. I am assuming that the son of Roger must have died before the earliest survivng parish register entries in 1558, although of course he may be one that was buried in 1586 if his son died young or moved to another parish. An entry could also simply be missing, but even if the elder Thomas was still alive when Francis was born in 1578, that would be at least 29 years since he was known to already have 3 children, so it is very unlikely he had another son that late.
Whichever Thomas was the father of Francis, it seems likely that he will be descended from the Byrds of Nether Locko, the last of whom was said to be William who sold that estate and bought Stanton Hall, but it is becoming clear that some of the family, possibly a junior branch, were already in the parish of Youlgreave before then. The two possible fathers of Francis appear to have been related as Roger Bryde mentions a William Bryde in his will, which it seems may be the one who owned Stanton Hall, whilst the William Byrdes who made his will in 1593, appears to mention another Thomas Byrdes besides his uncle. In neither case is any qualifier used to further describe the person, so that tends to indicate that they were both talking about someone in the same parish, and that there weren't multiple people with the same name.
The fact that Francis's will shows him to have been relatively wealthy seems to indicate that he might be more closely related to the main Bird branch, which owned Stanton Hall, meaning that Thomas the brother and uncle of the two Williams is more likely. However, there are two burials of Birds widows as yet unaccounted for in the Youlgreave registers, one is Anne in 1589, and one is Johanna in 1615. It is just possible that these were the wives of the two candidate Thomases. If that were the case (which is pure speculation of course), then given that they were both widows at the time of their deaths, Anne would have to be the widow of the grandson of Roger. As Francis had a daughter Anne, but did not have a daughter Johanna (or any variant), then that would make Thomas, the grandson of Roger, the more likely father of Francis. Both these claims, especially the second, are highly dubious, but illustrate that without further evidence it's going to be impossible to build a convincing case for which Thomas was Francis's father.
Brief details of his children:
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