A black market is simply illegal.  It could be that the goods themselves are illegal or it could be that the goods are associated with tariffs that the seller does not wish to pay.  Think of the black market as drug and arms trade.  Meanwhile, the GREY market (or parallel market if you prefer) is a market price set by the people and their perceived value of a good.  Let me illustrate by giving an example.  If the going currency exchange for dollars to yen is 1:1 and this rate is recognized by the government and exchanged at this rate at ATM's and banks and so forth, but the value as perceived by the population at-large is higher, your exchange rate could be 2:1, 3:1, or any other combination.  The grey market is far less seedy than the black market as it's kind of like a societal monetary control measure.  

There IS a grey economy which some people will recognize as being paid under the table in cash without having to pay state, local, or federal taxes and various social welfare programs.  This is not what a lot of people would call “illegal” so much as it's frowned upon, and THAT is where the grey part comes in.  There are no clear cut lines as to whether or not something is permissable so it goes on unchecked.  This generally isn't a problem with small sums of money, but if a CEO who made 7 million dollars per annum were to get paid entirely in cash, there would be one more corporate casualty.  

Believe it or not, many foreign economies thrive on the grey economy because they don't have NASDAQ, or the DOW, or the HSI telling them what their product is worth, but leaves it to the consumer.  Think of it like going to a market or bazaar and the price can be negotiated upon, or like a person to person auction.  This can maximize profits, but the price isn't always even across the board, but allows for fluctuation.