In this interaction, you are essentially a peer. You are offering input, but you have no responsibility for the success of the student who's material you are reviewing. That should influence how you decide to proceed.
People will receive critical review differently, and as you get experienced at this sort of thing (meaning, at the moment, as your mentor puts you in more and more of these situations) you will be able to tailor your behavior to meet the given situation.
At the moment, though, the person you are interacting with does not seem to welcome your input, and this is the issue that should probably influence your interaction the most.
Without bothering to try to figure out if your input is constructive, valuable, of the appropriate tone, etc., it's very difficult to advise you. However, it is unlikely that this person's attitude toward your contributions in review will change -- certainly if you don't change your input style, and probably not even if you do. The relationship has been drawn by your history.
Take a step back, and realize that there are many roads to a successful conclusion of a writing or research task. I'd recommend, given your situation, holding your input to instances that you feel are truly important -- gross errors, flaws in logic that will result in a wrong conclusion, and the like. Try to avoid picking nits. There's a chance that if you back off of the little stuff, your colleague will realize that your input is valuable, and adjust the attitude accordingly. If not, and you're facing unpleasantness anyway, at least you're doing it for the important stuff.
Also, try not to limit your comments to stuff that needs work. Try to highlight whatever strengths you see in whatever you're reviewing. Many grant reviews, for example, open with a section on "strengths", followed by a section on "weaknesses" Forcing yourself to focus on strengths first will make you weight weaknesses more appropriately. Sometimes you just forget how much telling your colleagues "good job" can mean to them. When it's largely a good job, don't be afraid to say so, loudly and publicly.
Lastly, there is such a thing as "new reviewer syndrome". It's horrible when it effects who gets funding. If you tell a room a whole list of really nitpicky criticisms, and then try to give that grant a high score because the criticisms are not particularly important ones, that can really kill a grant.
Given your comment, it's clear that you actually have some responsibility. Then, you are what's referred to as an "Accidental Manager" in a great book by Heerkens: https://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=9780071818520 -- you have responsibility, but no authority over your team, which might be (a) peer. You can't hire or fire, offer bonuses, provide input into their employment decisions or actions, punish, etc. I recommend taking a peek at that book. It's a good read, and might provide you with some better insight to your situation.
The advice offered above still holds, but there are some additional management issues you should consider. Actively think about your management style during your interactions. http://vault.theleadershiphub.com/blogs/conflict-styles presents cartoons of some conflict resolution styles. Before people overly criticize -- these are JUST cartoons, but they raise some interesting points.
For the interaction you seem to be in, I'd recommend avoiding the styles where things need to proceed your way, sacrificing for the good of the overall project. The person you're working with might feel like they're at the losing end of a win-lose proposition. Thus, you probably need to avoid being the Shark. If Shark is called for, it's probably you're boss's place to do it. You don't want to be walked all over, either, so Turtle is probably wrong.
Just as a word of caution, I'm still not sure of your absolute status. If you're a Master's student in a lab, "I have a responsibility to my supervisor to make sure the work done in the lab meets certain standards" does not quite match your role. You have a responsibility to make sure your work in the lab meets certain standards. If you try to make sure that somebody else's work meets some standard, you might well be overstepping your role, and that might be contributing to the reticence to accept your input.