You aren't entirely correct in what you are saying here. If Company A owns Company B, then it is a subsidiary and an affiliate of A. Owning a share of Company A gives you an indirect ownership interest in Company B (which is simply an asset of Company A). As a shareholder of Company A, you can, proportionally control its actions (through votes; board representation). And Company A as the sole or controlling shareholder of Company B, can control its actions.
As for the liability issue, you are correct that as a shareholder of a major public corporation like BP, the scope of your liability would be limited to the extent of your shares (the hallmark feature of a corporation (limited liability)). However, Tailwind was not a publicly traded corporation. It was likely a closely held corporation and liability would be more meaningfully distributed (and felt) by the shareholders of a closely held corporation typically also RUN the corporation.
Now the twist here is that someone one was managing the Company, and someone was separately managing the team and its equipment. In the latter case, that is clearly Johan Bruyneel (formally) and Lance Armstrong (informally). The link is that Tailwind hired Bruyneel and Lance, and as such is responsible for the actions of its employees. So even though Weisel and Tailwind's other investors may have had no actual knowledge of Bruyneel's alleged sale of team equipment to fund an alleged systematic doping program, Tailwind is responsible for any liability that resulted from those actions.
At least that's my initial read on this situation.