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.