What is the idea of parliamentary sovereignty from the American and British point of view?

Strictly speaking, the UK has parliamentary sovereignty,
but the United States does not.  This is due to the technical definition of this
concept.  Parliamentary sovereignty occurs when the Parliament or legislature is the
supreme legal body in the country.  It can, essentially, create any law it likes and
change any law it likes.  No other body can override its
will.

In the UK, as you can see in the link, Parliament is
actually sovereign.  There is no independent executive branch that can veto its actions
and no judicial branch that can rule them
unconstitutional.

In the US, Congress is not sovereign in
this same way.  The President can veto laws passed by Congress (though Congress may
overrule the veto).  The Supreme Court may strike down laws passed by Congress.  These
other parts of government have a check on Congress.  The US Constitution is set up this
way because the people who wrote it did not trust any one body of government -- they
wanted each body to have the power to check and balance all the
others.