There is an oft quoted research about the history and descent of the Lindsays in Scotland that is very much worth reading.
Since I have not been able to get further back than about early 1800's on our main line I cannot "make the leap" so to speak. William's parents MIGHT be James Lindsay and Agnes Loban but that has not been (concretely) confirmed as far as I know.
Another interesting read and a wee bit different information wise can be found at http://www.electricscotland/com/webclans/htol/lindsay2.html
The old Scottish chronicler Andro is oft quoted as saying - "Out of Englande came that Lyndsay; Mair of thame I can nocht say." (and sometimes I feel the same way about my own!!!)