Answer:
You can see Mozart's manuscript, for example concerto in C major:
http://www.themorgan.org/collections/collections.asp?id=185
(copy/paste the link into browser)
Besides strings (violins, violas, cellos and double basses), there are "2 oboes, 2 horns, 2 bassoons" etc. A small orchestra. And it's clearly written "Cembalo". I believe it means "harpsichord", not piano.
Cembalo does mean harpsichord, but the harpsichord and piano were used almost interchangeably in Mozart's day.
Here is a good example how Mozart's concertos should be performed:
http://www.youtube.com/watch?v=9ACFvHCrlXo