Did you give the bare wood a lye wash before you tried staining it further?
Lye water will dissolve the tannin in the wood and float it to the surface. The areas where the grain is running parallel to the surface won't show much difference but, the areas where the grain is breaking out of the surface (think, "endgrain") will release quite a lot of tannin.
That will cause the stripes to appear much darker than the adjacent areas.
To use lye water, mix about a teaspoon of dry lye into a cup of water. Use a small paint brush made with synthetic bristles. This is important because the lye water will dissolve real hair bristles leaving you with nothing but a handle.
After whiskering the bare wood, paint on one very wet coat of the lye water and let it dry.
Following this, paint a wet coat of vinegar on the wood to neutralize the lye. Then, proceed with whatever staining you want to do.
Needless to say, don't get even a tiny drop of that lye water on you. Especially in your eyes. It can blind you and it will dissolve any flesh it contacts.
Knowing this, I always have a bowl of vinegar standing by so I can neutralize the lye. That said, I can't think of one time when I was doing this that at least one drop of lye water didn't get on me somewhere. Instant fire in a tiny spot is the best description I can think of to describe the feeling.
Applying more than one wet layer of lye water didn't seem to make the contrast between the light and dark stripes for me. It might for you but my suggestion is not to bother with trying it.
When you finish with the lye water, dump the rest of it down your bathroom sink. Any eyelashes, beard or other hair that went down the drain will be dissolved.