Best 'private' spots for stargazing on Mountain Teide, Tenerife

I'm planning travel to Tenerife in the end of May and one of the things which I like to do is do enjoy in a beautiful star scenery. I did some investigation and I concluded the best place is somewhere around the mountain Teide which is far away from resorts and therefore lights. I'll rent a car so getting there is not a problem at all. The real question is, are there any good places around or on Mountain Teide where I can drive on the night and just enjoy the stars (without telescope or similar, I'm pretty sure that I'll be shocked with my eyes also because I never saw too many stars, damn light polluting).

So I'm looking for good spotting places with following:

1. If possible private spots, because I'm planning to go there with my girlfriend and want to have as much romantic mood as I can

2. Reachable by car, because I don't find it relaxing walking too much outside of the road in the middle of the night in the middle of the mountain :) I can do some walking, but not too much :)

I saw that there are many stargazing tours, but I'm looking something private for couples where we can just sit somewhere and enjoy the stars without dinner or similar experience which are offered.

Any other advice is welcome and if you have any suggestions just let me know. If you can reference spots on google maps or similar, that would be perfect :)