Using Safari, I browse to google.com. When I type in the search field, no text is displayed. If I resize the window, previously typed text appears. The issue is not seen with macOS 12.2.1. The issue is not exclusive to Safari and does not reproduce outside of VM.
I don't have a solution but I am seeing similar problems with 12.3.1. I use a vm to run Apache - MySQL - Wordpress to develop WordPress sites. I'm using a MacBook Pro running Monterey 12.3.1, and have no problems with a vm running 12.2.1. However, after I upgraded the vm to 12.3.1, I noticed when I entered data into fields in Safari, they would stay blank, though the data was entered because I could submit say a login screen and it would log into a WordPress site running on the vm. Other problems I noticed is that scroll bars in Safari on the vm would not work, or would work very slowly. The same with clicking checkboxes. All of this worked fine with 12.2.1. So I've gone back to a vm running 12.2.1 as it works and is fast. So you're not the only one noticing that there are problems using 12.3.1 in a vm. I did notice that when I updated 12.2.1 to 12.3.1 in the vm, that it took a handful of reboots before the vm would boot. Two times, the vm just quit partway during booting after updating it to 12.3.1 and I had to restart the vm.
I have described the same issue you are having in the post linked below. TLDR: The graphics hardware acceleration in the most recent Parallels version (v17.1.2) is broken. https://forum.parallels.com/threads...y-bug-in-parallels-17-1-1.356945/#post-903121 My advice is to download Chrome and disable hardware acceleration in it (Safari does not support disabling hardware acceleration). I haven't seen this issue getting any attention from the Parallels developers so far - no posts on these forums acknowledging this as an issue or whether they are looking into it.
