To address your concern directly; No. It will not harm your tires. Assuming these are traditional construction grade stones, they will not harm the tires. If they are unrefined stones collected by hand there could be a concern that the type of stone could have sharp edges, but this is unlikely. 3/4" gravel, traditional for semi-improved parking spaces is generally tumbled for grading purposes. This results in relatively rounded gravel which is unlikely to cause a puncture at the low speeds of parking.
Another option is to bury some cinder blocks and park on those. I have done this for my front yard, for when I need extra parking. Just make sure that the hole dug for the block to sit in is clear of rocks, and putting down loam and/or sand bed to prevent them from cracking is a good idea. I failed to do this and one of my blocks cracked, but all the others are intact.
The only remote concern is the increased level of humidity released by the dirt vs pavement.
If you plan to move the vehicle regularly, then this is less of a concern. A lot of other factors come into play here, as well. If there's a lack of plants to come into contact with the bottom of the vehicle? What height is the vehicle; Obviously there will be more air flow on something with a decent amount of ground clearance vs a typical "sports car." etc.
If you are concerned about the additional moisture released by soil/gravel parking, you could put down some manner of vapor barrier, like a heavy mil Plastic/Tarpaulin, and park over that.
I actually saw your post the other day when I was researching this, as I tend to keep going back to the topic. There does not seem to be any science in the way of making the decision concrete. Common sense tells us that pavement will release less moisture, but we can't imply that fact makes for measurable difference in undercarriage corrosion.
One thing that has more pros than cons would be to have the vehicle undercoated. This would help reduce corrosion to the body, and some what mitigate the excessive moisture of "semi-improved" parking spaces.
I am sorry I do not have any science for you, but I have searched on numerous occasion and found only conjecture.
I hope this helps in some way.