There is no harm in acquiring knowledge which would be useful both in acquiring an income and in caring for a home and family. Of course, it depends on you own inclinations and gifts of which you say nothing.
But just a stab in the dark, here are some things which would be most useful and feminine, and not necessarily be in competition with men:
Nursing, teaching, naturopathy, herbalism, gardening (knowledge and practice), podiatry, occupational therapy, floristry, home decor, dressmaking, tailoring, child care, cookery.
Are your parents actually trying to push you out or would they be satisfied if you were to decide on a path and take it?
All of these require a college degree if one is to do them to the level of self support.
Nursing and occupational therapy both require a lot of schooling. You can't get into OT without a graduate degree. Ain't cheap. Some upper-level nursing jobs require a Master's, too.
I don't see any problem with a lady doing men's work - as long as she does it modestly. I don't see why that would be impossible.
I agree. At the end of the day, it comes down to whoever's most qualified for the job. Here's an example:
People with a master's degree in my major go on to lead college and professional choirs. If the most qualified candidate for one of those jobs is a well-rounded, God-fearing woman (most of the country's best-known college ensembles were originally church choirs and many do still perform religious music) with decorum, then she's the most qualified candidate.