Ok well first off, Saturn won't be transiting Virgo for another 20-something years so you don't have to worry about that.
For family and reconciling relationships, you will need good Moon and Venus energies.
Venus is currently retrograde, and Venus rules peace and harmony etc, so this is not a good time yet. Btw Mars retrograde is good for a working like this but it won't be retrograde for a while yet.
Venus will be in Libra (a very good placement) in November. On November 29th, Moon will be in Cancer, and day of Sunday so that would be a pretty good time to start this working. If you wanted to start the working sooner, wait until Venus is direct again, in late September it will be direct in Leo so a decent time. October 4th and 5th, Venus will be in Leo, Sun will be in Libra (good), Moon in Cancer. Day of Sun (4th) or day of Moon (5th).
Also... make sure you 100% want your relationship with your parents to be mended. Workings don't work out if we are not 100% in tune with ourselves and the working. Speaking from personal experience, sometimes we are better off without some people in our lives.
Hail Satan!