Why is the president not allowed in Parliament?

The USA has a president but not a Parliament. England (or rather the United Kingdom) has a Parliament but not a President.

The chief executive of a nation is not allowed into the legislature to ensure the checks and balances and separation of powers of a country.

The UK has a Prime Minister, and he (or she - only one so far!) is also an elected constituency MP, and does participate in debates in the House of Commons.