Mangalik dosha is always checked from natal chart as well as moon chart. In the birth chart of boy we see that Mars is in the 7th house of natal chart and 9th house of moon chart, hence boy has partial mangal dosha.
In the horoscope of the girl we see that Mars is in the 5th house of natal chart and 8th house of moon chart. Hence the girl has partial mangal dosha.
As both charts have partial mangal dosha they are compatible.
Why your parents are not agreeing to your marriage, it is because of Mangal dosha or it is because they are against love marriage. Or they think that you are still young and cannot make a decision for yourself.