It is true that the stronger gravity of the Earth would tend to make a spaceship fall faster, while its thicker atmosphere would tend to make it fall more slowly. Which of these effects proved stronger would depend upon the design of the parts of the spaceship designed to make the atmosphere slow it down. A very small spaceship with a very big parachute would fall more slowly on Earth, because the air drag would dominate the force of gravity, while a big spaceship with a small parachute (admittedly not a great design) would fall faster because the small parachute would not be able to dominate the increased gravitational force.

The atmosphere on Mars is MUCH thinner than the earth's but the planet itself isn't much smaller than earth.Without the parachute open, the lander would accelerate faster due to the earth's size and have a slower terminal velocity due to the denser atmosphere. With the parachute open the terminal velocity would be MUCH slower because that is what parachutes do--decrease your terminal velocity. Since the parachute is designed for a thinner atmosphere, would it be bigger or smaller than if it were designed for a thicker atmosphere?

This is a very good question which involves the balancing of forces. The force of gravity is greater on Earth because it is a bigger planet but the atmosphere on Earth is also much thicker than on Mars. While gravity tries to speed up the object towards the surface, air resistance slows that speeding until the object falls at "terminal velocity", the velocity at which the desire of gravity to accelerate an object towards the center of the planet is completely balanced by air resistance slowing the object down. The air resistance is determined partly from the area that is perpendicular to the direction of fall as well as the density of the air.
Parachutes use this idea because they effectively increase a person (or Martian Lander)'s area perpendicular to gravity and thereby slow down the fall. Some problem with parachutes are that parachutes need enough air density to be effective, they can only withstand the force of slowing the object for a limited time before ripping under the stress and also winds in the atmosphere such as the jet stream might affect a parachute. For these reasons, parachutes usually aren't put out until the object has fallen through the atmosphere for a while. What other mechanism do you think might work to slow down an object in the atmosphere? Although the gravity of Earth is greater than Mars, I think that the atmosphere is so much more dense near the surface that the lander would be falling more slowly on Earth but I can't be sure that's the case because I don't have all of the numbers to do the calculation here. The scientists and engineers at NASA would have calculated these figures very exactly to make certain that the lander would be ok. On the lander, they used airbags to cushion the last part of the landing. What sorts of methods do they use for objects returning from space to Earth to slow the final descent? Think of the Space Shuttle and the Apollo capsules.

One thing that I think is important to discuss is that the Mars Pathfinder landing was very carefully designed for the conditions on Mars. The characteristics of the atmosphere and gravity were taken into account when deciding on the necessary trajectory of the spacecraft and how to build it. The spacecraft was very carefully designed so that the heat shield had the right size and shape, the parachute was the right size, the retro rockets had the right amount of thrust and fuel, and the balloons could handle the impact. Also, the landing procedure was worked out so that the various landing phases happened at just the right time. The parachute had to open at the right time and stay attached to the lander for the right duration; the rockets had to fire at the right time and stop at the right time.

I guess the point I am trying to make is that the engineers at the Jet Propulsion Lab made quite an effort to figure out how to land Pathfinder safely on Mars. This was also an experiment. The main goal was to get a lightweight lander to Mars very cheaply (relatively). They could have used bigger rockets to do a soft landing but would have meant more weight and expense and maybe more time to get to Mars.

So as far as landing a similar spacecraft on Earth goes, I would argue that it would have been done differently to take into account the differences between the Earth and Mars. I don't really know offhand what would have to be changed but it is possible that the Pathfinder as designed could not have landed safely on Earth.

for more info about Pathfinder you can look at the JPL website:
http://mars.jpl.gov/MFP. One thing I found almost right away was a picture of the landing sequence.