The question I have is "what internet connection was be using?"
If Wifi, then why does the Houses of Parliament not have filters to stop access to unethical material (such as Porn), most work places would. However, even if they don't given that it's possible for you to be found out then if you accidentally went to the wrong site the thing to do is instantly close the site and go and confess that it was an error.
In this case such a confession would have been at most 5 minutes after the story broke, if not before. Yes it would be embarrassing, however most people would accept an error.
As such this would be the timeline; wrong site opened and closed within (say) 20 seconds (i.e. long enough for someone else to observe the site was open), then email sent to chief whip explaining that a site was visited in error, news story breaks, chief whip asked for comment, comment is "the MP in question has already contacted me, explained that they opened a similar sounding site to the one they ment to visit and closed it as soon as they noticed their error, as such we don't believe that further action against the individual is required at this time", a few people get upset but far far fewer than is currently the case, chief whip confirms that the site has been added to the banned list of sites able to be visited on the Houses of Parliament WiFi.
Now I know that generic filters can be a bit over enthusiastic (for instance our home broadband provider makes it hard to even visit sites selling underware, much to my wife's frustration when looking to buy new underware), however a workplace with an IT department should be able to better manage such issues (and probably isn't trying to set the limit for children). As such it should be fairly easy to allow sites, so no work is impacted, but also stop the most obviously pornographic sites from being accessed by accident.