Yes, this is a common, well known problem with parts that are being held in milling vises. The first thing you need to do is understand just what is going on to produce this lift. I made this sketch some years ago to show what is actually happening.
Only the important parts of the vise are shown and I do not show the parallels that are almost always a part of this situation. The dark gray at the bottom represents the bed of the vise. The lighter gray block on the right is the fixed jaw and the one on the left is the movable jaw. The vise's screw is shown as a white rectangle on the left. And the white block between the jaws is the item being worked on: it is shown in an elevated position as the parallels would place it.
The top illustration shows the situation just before the vise screw starts to apply any pressure on the item being held. Notice that all the parts are square and parallel to each other.
But when the screw starts to apply pressure, as shown (somewhat exaggerated) in the bottom, things start to react to the uneven forces. The item being held stops the advance of the movable jaw at the top, but there is nothing at the bottom to match this. So the bottom of the movable jaw continues to move for whatever distance is allowed by the looseness of the fit of it to the vise bed. There must be some amount of play in that fit or the movable jaw would not be able to move.
The left, bottom corner of the movable jaw becomes a pivot point and the movable jaw rotates about it. This lifts the right, bottom corner of that jaw so it is now above the vise bed and at an angle to the vertical. This, in turn lifts the left edge of the item being held while the fixed jaw, which does not move, holds the right side of that item at it's original height. So the item being held is both lifted, more on the left or movable jaw side and tilted. This lifts it off the parallels.
It is important to notice that this effect mostly happens when a part is supported on parallels or some other means, ABOVE the level of the vise's screw. When a part goes below that level or when it rests on the bed of the vise, the effect is very minimal and normally no corrective action is needed.
What can be done about it?
There are high end milling vises that employ a system of splitting the force of the screw into a combination of vectors: toward the fixed jaw and DOWN, in roughly equal parts. This tends to limit the lifting action. Also, high end milling vises are built to tighter tolerances. However, neither of these features nor the two combined will completely eliminate it. So far, this is the best system available.
There is a style of vise that is called "screwless" or a "machinist's" vise. These vises are not actually without a screw: the screw is there but a lot less visible than the normal vise screw which is in a horizontal position. In a screwless or machinist's vise, the screw is INSIDE the movable jaw and at a 45 degree angle in a downward direction. So as the screw is turned it both moves the movable jaw toward the fixed jaw and pulls it down. These screws have a pivot at the bottom end, are somewhat loose in their hole in the movable jaw, and have a limited range of action. So there are a series of positions where the bottom end of the screw can be placed so that the vise has a longer range of opening. This style of vise is also ground on all six sides to provide precise alignment on a milling table or in the jaws of another vise. I have two vises of this style and they work well.
Some, less expensive vises use a separate guide in the form of a horizontal rod that is fixed on it's two ends and which passes through a close fitting hole in the movable jaw. This also helps to prevent the tilting but is not as effective as the force split of the high end vises.
Another method that many vises use is to make the movable jaw longer and to clamp it on bottom facing ways on the bottom of the vise bed. Again, this helps but is not as effective as the above features (at lease in my humble opinion).
Tricks:
One trick is to use a DEAD BLOW hammer to tap the piece down as the vise's screw is being tightened. This dead blow hammer should have a soft face so that it does not damage the part. This is perhaps the most often used trick.
Another trick that I have used is to place a second piece of stock below the one on the parallels. This second piece of stock is sized to hold the bottom of the jaws apart by the same amount that the desired part does at the top. If it is accurate enough in thickness, the parallels can sit on it. Also an adjustable parallel can be used between the two main parallels that sit on the vise bed.
BUT, one of the best "work-arounds" that I have seen was in an internet video. The rough stock was placed on parallels with only a small part of it's bottom being gripped. Most of it was sticking out, above the top of the jaws. Then ALL the operations could be performed on FIVE sides to square it up and create any holes or slots or interior openings in the final part. All of this work was done on the part of the stock that was ABOVE the vise jaw tops. And, with some care in the direction of the cuts, these five sides automatically come out as square to each other as your mill allows. Then, the almost complete part was taken off the parallels and inverted. It was set as low as possible in the vise jaws, preferable against the vise bed. And then the sixth side was cut down to below the area that was originally used for holding it on the parallels. Ta, DA! Of course, this only works on parts of certain sizes and there is some waste when that sixth side is milled down.
Once you know the problem, there are many ways to work around it.
PS: Do not allow thermite to discourage you.
This forum is oriented to the professional machinists and they often have limited patience toward new-bees.
In line with that, you may want to join another, home/hobby oriented machinist BB. A good one is:
General -
The Home Shop Machinist & Machinist's Workshop Magazine's BBS
Many people here also visit there.