Hitler had no policy of avoiding the bombing of historical sites. A walk around central London will take you past any number of churches that were destroyed or heavily damaged in the Blitz. Many of the historical sites you mention took some bomb damage.

The big reason that landmarks like the Houses of Parliament and Buckingham Palace survived is simple geography. The prime target of the bombing was the Docklands of East London. This is a historically poor and industrial area, with few major landmarks. West London had few strategic targets and as such was only hit by scattered bombs, but also has more landmarks. Had Whitehall been hit as hard as Stepney, it's hard to see the landmarks surviving.

For those landmarks that were closer to the bombed areas, they were saved by massive efforts on the part of the emergency services. On the night of the 29th December 1940, St Paul's was hit by several incendiary bombs. The area around the cathedral was burnt to the ground. The London fire brigade and a volunteer fire watch service dedicated their efforts to saving the cathedral, and prevented serious damage. Part of the House of Commons was destroyed in a later attack, but firefighters prevented the fire from spreading to the rest of the building. Many of London's major landmarks have similar stories.

Some survived through sheer luck. In January 1940, Bank underground station was hit by a German bomb. Had this hit a few tens of meters northwards, it would have destroyed the Royal Exchange, while if it had hit a similar distance to the northwest, the Bank of England would have been destroyed.

Both Buckingham Palace and St Pauls were damaged during the war while other significant buildings like Christ Church Greyfriars and the Carlton Club were destroyed.

http://www.bombsight.org/ shows where each bomb landed. There is clearly no evidence of intent to avoid hitting any significant landmarks, and given the state of radio navigation and bombing accuracy at the time, its highly unlikely that any such policy would be a reasonable proposition anyway.

Of course, damaged buildings that weren't demolished were repaired, however, bomb damage is still evident on a lot of buildings that were extant in the the early 1940s, some of which has been memorialised, some of which is just... there...

This video might be of interest to you.

http://www.nam.ac.uk/whats-on/lunchtime-lectures/video-archive/scars-war