Do your parents ever give any money to charity? To their church, to public TV, to the animal shelter, the Red Cross, etc?
Transfer the shares to the charity instead of making a cash donation. Don’t sell the shares first, just transfer the ownership of the shares.
The advantages of this are: you don’t have to pay tax on the gains, you can take a deduction for the current fair market value of the shares, and neither you nor the charity will need to know what the basis was. It won’t cost you anything (except possibly a fee to transfer the shares) since it will be replacing a cash donation you would have made anyway.
Note: This trick only works with long-term capital gains property, which is generally property you have owned for more than one year or that you have inherited.