An easy fix would be to put the transparent parts in the alpha channel in Photoshop and save it as a .psd or .tga and import that instead. .png can act a bit wonky.
You can turn on/off viewing the alpha at the top left corner of the viewport when opening a texture and you need to plug in the alpha channel in opacity mask in the material editor to make the material use it.
Here is another thread started by someone with similar problems. It has some useful replies you can read as well.
When you import a PNG texture with transparency it can’t use it as actually transparent, instead it has to create an alpha channel of the transparency. That means that your RGB channels have areas without any pixel information, so it will look at the pixels that are there and try to extrapolate to the rest of the image. If it replaced it with some other color–like black–then it would have a black color where the transparency is rather than a nice transparency on your texture.
Other formats like TGA require an alpha channel so you set the colors yourself.
That works but you could put the black/white alpha in the alpha channel of your original texture to save texture sheets It’s more optimized that way but would require the texture to be saved as .tga or .psd