They do ship discreetly (they don't say Diapers on the box or anything obvious like that), as far as thickest; the Overnights are their best, though one of the newer ones supposedly is 500ml (.5 liter) better.
Tykables sucks in 1 way though, they never seem to actually be in stock and so if you are trying to get a shipment at a convenient date that is basically impossible; for me this is a much more important thing than whether or not the package actually says diapers on it.
In this way I have found ABUniverse much better, of I order on Wednesdays or Thursdays and they say that they are in stock at their location closest to my house I get my shipment by Friday or Saturday (so I can sneak a shipment in the house and not get asked about it)