I'm so sorry for your loss, you've been through a really rough time and it's not surprising you feel all over the place emotionally
. Of course you feel really attached to the house, you have so many happy memories there and it feels like a connection to your loved ones. Don't ever feel stupid or wrong for feeling that way.
I would just echo what some of the other kind posters have said, that it's very normal to feel a lot of nostalgia and melancholy about your childhood home (or other places that have a lot of emotions attached to them) if you can no longer go back there, but what you are actually feeling sad about for the most part is that you can't go back to that time and to those people you were happy with, not usually for the place or the house itself. That is what is so sad about bereavement, you can't ever turn back the clock and put things back to how they were no matter how closely you recreate the physical conditions of how things were...
Physical objects can bring back a lot of memories and evoke powerful feelings but they aren't actually the same as the feelings themselves, you can not have the object but still have the memories, if you see what I mean? So although I know what you mean when you say the house is your last link to the area, it really isn't, you have a really powerful link to the area in the form of your memories of growing up there and all the other happy memories of your parents. Nothing can take that away.
For that reason I think personally I would sell the house, unless you do have a lifelong dream to live in your parents area and selling the house closes that option off, which doesn't sound like the case really? If you do decide to move back at a later point you will have the money from the sale of the house to do so? I definitely wouldn't rent it out unless you are in dire need of the money, for the reasons others have said, being a landlord entails a lot of expense and hassle and even an above average tenant will probably not look after the place as if it was their own, and things like the garden will probably get neglected or worse. Plus you will still have to deal with all the emotional consequences of 'letting go' of the house without any of the upside.
I know it's so hard to let go and I felt exactly the same with my parents house, I was desperately doing sums and trying to make it work for us to live there - I even persuaded manipulated my DH who absolutely loves where we live now that it would be worth it
but I'm glad we didn't in the end. It was 100% grief talking and not me rationally thinking it would be a good thing for me or the family, had that been the case we would have moved back to the area 10 years before. Obviously each person is different but I really think when you are grieving a major loss is not the time to be making major, life changing decisions.
I honestly found it a bit easier with my parents place once all the 'stuff' was out of the house - once it was stripped of so many of the things which I remembered it looked different and I found it easier to think of it as 'just' bricks and mortar. Also it looked really quite forlorn and unloved as most houses do when empty, and although that was said I could then be happy (ish!) that someone new was coming in to look after it and make it a home again. It helped that we sold it to a nice young couple who were planning on making it a long term home (or so they said, I chose to believe them anyway). I did do a bit of a cheesy thing of walking around saying 'goodbye and thank you' to the house which although unusual for me (I don't usually talk to inanimate objects!) made me feel quite a lot more at peace with letting it go. You don't necessarily have to sort through all your parents possessions/stuff in one go, I think in many ways that is the really hard thing emotionally, I know some people are dead against this but if you need more time could you rent a storage unit near you and put everything in there, then you can sort through at your own pace and do as others have said, take lots of pictures of the things you can't keep and find good homes for things that still have life/use left in them?