You can add the relevant Taxonomy parent Field to your Views, make it display the term ID. This way you get that information available as a token.
You can hide that field with Exclude from display
and position it above all the other fields, so you can use it as a token in all of them without worrying about the position.
Then edit the Style settings
of your "Parent" field that you want to style and enable the Create a CSS class
option. You can enter the token for your parent TID in there.
Since css classes can't start with numbers make sure to enter a safe start for the class, for example: myclass-{{ tid }}
To find the list of tokens you'll have to temporarily enable the option Rewrite results > Override the output of this field with custom text > Replacement patterns
.
You may need to Customize field and label wrapper HTML
and/or Use field template
depending on your particular configuration and output to get the class to show up.
Alternatively, you can construct the output of your field on your own with the Override the output of this field with custom text
and enter what you want the HTML to show:
<div class="myclass-{{ tid }}">{{ field_output }}</div>