To know how to treat the spots, you need to find out what they are. If they were powdery then I'd suspect mildew - the cure would be to keep the soil evenly moist and ensure the atmosphere isn't too damp. If they're thin patches then scorch seems possible. To cure that, ensure the foliage doesn't get concentrated light directed at it. There are many other possibilities.

I can't think of any good that vinegar and baking soda could do for a plant, so I'd suggest you avoid that in future. If you're going to spray a plant, it's important to use the right product or you'll do more harm than good. If you don't want to use the correct commercial product it might be better not to spray at all.

You must be doing something right to grow parsley on a windowsill! The spots may be from sunlight hitting wet spots on the leaves and burning them; or it could be mildew, in which case better air circulation would help; and if it's the result of a bug, shaking the leaf over a piece of white paper will likely tell you if you have "guests."
Vinegar is an herbicide, so I wouldn't use that. A solution of baking soda and water is helpful in preventing damping off, but that isn't your problem. If the problem persists, put a couple of affected leaves in a sealed plastic bag and take it to a nursery. Someone there may be able to identify the problem.

Parsley does very well for me here in the subtropics, even in the full blast of the afternoon sun. But I wouldn't be tempted to put it near the hot glass of a window. Plants' leaves can burn very easily that way. It's possible that the white spots were the first signs of scorching. However, because most problems with inside plants are due to fungal infections or insects, one of those is the most likely cause.

Baking soda (bicarbonate of soda) has its uses in the garden, but can be damaging if overdosed. It will help to raise the alkalinity of soil which is a bit on the acid side (good for tomatoes in such cases); it can act as an antifungal when mixed in the proportions of 3 teaspoons baking soda to 4 litres (1 gallon) of water (NOT vinegar!! and best applied as a preventative, less effective as a treatment. It can burn the leaves of susceptible plants, so test on a small area first. Also, don't use the mixture to water your plants - the soda can interfere with the absorption of water by the roots.); you can use baking soda to test the pH of your soil (add a pinch of baking soda to 1 tablespoon of soil. If it fizzes, the soil's pH level is probably less than 5.0).
The first thing to try is chamomile tea. Just spray it on - it's a specific for fungal infections on plants. If that isn't effective, opt for a garlic spray - that also is an excellent antifungal treatment, but it will also deter a lot of insects. AND it's safe for plants and humans. Your parsley won't taste of garlic once you've given it a rinse in water.

Last but not least - no plant was designed to grow in a house. It is anything but an ideal place for them, and they'll all benefit from a little time in the great outdoors, where they'll get the REAL unfiltered rays from the sun, and where predator bugs can deal with infestations.

A lot of people on the organic gardening forum say they use vinegar to kill weeds. So the vinegar probably killed it.

I've read the same organic advice for using vinegar as a weed-killer. I can't imagine vinegar even mixed with baking soda would be beneficial at all for a plant.

In the garden, parsley can easily bounce back from being cut to the ground but in the house...dunno. Parsley was never meant to grows indoors so it isn't as robust there.

FataMorgana

I too have heard vinegar is a weed killer.

JoJo