What do you think is Amethyst Star/ Sparklers relationship to Dinky and Derpy. Sister? Half-sister? An adopted sibling?

My take on it is that Sparkler is a part of a 'Big Sibling' programe with Dinky.

Dinky is Derpy's daughter, and that's the extent of their family. Amethyst Star is completely unrelated to them, save for being a good friend and occasionally babysitting Dinky.

Time Turner has absolutely nothing to do with any of the three and can go die in a fire.

4293785
Well, in the Sisterhoof Social episode, Amethyst did the race with Dinky. There were three main pairings they showed during that race: Berry Punch and Pina Colada, Amethyst Star and Dinky, and Applejack and Sweetie Belle. Either their sisters or it's ridiculously common for non-sister pairings to compete in it.

However, there is no evidence in the show itself that Derpy is related to either one of them. In fact, the adult we see Dinky with most often in the show is Carrot Top.

Also, Goldenscript was shown nuzzling Carrot Top, and with both Carrot Top and Dinky before.

It would seem logical to assume that the family unit is Goldenscript, Carrot Top, Amethyst Star, and Dinky Hooves. Though since they're all background ponies, no one is going to hold you to this and you can still do whatever you want with them.

4293785 I tend to portray Amethyst as being Dinky's (much) older sister, and their parents being absent/dead, so she, despite being only a young mare, has to raise her younger sister.

I haven't really looked at the Goldenscript/Carrot Top side of things, but yes, they are often seen with Dinky. I'd work that into my headcanon by saying they're perhaps Amethyst and Dinky's aunt and uncle, or some very old friends of their parents, that acted as surrogate parents for Dinky and Amethyst when they were younger, and still look after Dinky when Amethyst is busy with work. (She's a jeweller in my mind.)

I'm so late to comment my thoughts, I actually think she don't relate to any of them. I think that Amethyst is Dinky's foalsitter, and some other foals' ,too. She mostly foalsitting for Dinky and Rumble (and I ship it), and helps Thunderlane with his house chores (and I also ship it).