I'll leave it to others to answer regarding the parking but I would use the Foggy Bottoms Metro stop (the closest) or preferably the Arlington Cemetery Metro stop (more scenic). From the Arlington stop you would walk across the Memorial Bridge right into the back of the Lincoln Memorial (and around it to the Reflecting Pool). If at all possible go at night (though the Arlington stop does not operate at night).

One of my most memorable DC moments was a bitter night in November 2001 when I went down to the monuments and was essentially the only person there.