I am often a fan of a gritty campaign that offers deep personal challenges to the players. For example, my Druids really struggled with the unliving gray stone of our urban campaign. We made the most of the role play and enjoyed our weaknesses as much as our strengths.
In such a campaign, character weaknesses and flaws are to be embraced rather than avoided. They add depth. Struggling against the societal injustices that are inevitable when trying to live as a hated Drow in a city of sun dwellers can be intense and thrilling.
But what of lighter-hearted campaigns? What if you just want to be dark-skinned? Or what if you just want the abilities of a Drow without the societal baggage?
There's a few things you could consider.
You could choose the stats of another eleven race and just be black-skinned, in the same way that humans can be black-skinned.
You can craft a new racial subtype. The DMG gives guidelines for this. If your DM is comfortable with crafting a custom Drow, say, one that grew up on the surface, he may let you trade some benefits, such as dark vision, for eliminating the drawback of light sensitivity.
You can also discuss with your DM the idea of relaxing the rules modestly. "Direct sunlight," for example, might be house-ruled to exclude early morning and late evening sunlight, when the light is arguably less direct. He might also allow the partial shade afforded by trees and large buildings to shelter you. Only consider this option if your campaign is generally lighthearted and your DM is generous and open to customization.
I find the idea of a donkey with a parasol amusing. If your campaign is very comedic, stick with that, as it was your idea to begin with, and has endless potential for amusement. My mind is now drifting off to all the donkey jokes in Shrek.