Lived with partners' parents' for 1 yr, but don't know how to prove?

Don't include affidavits or letters. They won't make any difference.

Unfortunately, letters from friends and family will not have much standing as evidence.  And I'm afraid that 7 months is quite lengthy; if you can't find official evidence to cover that period, I doubt the visa application will be successful.

For a fiance(e) or spouse visa, there is no need to have lived together.  So for these visas, a friend or family member can provide accommodation and write a letter confirming the applicant will have an appropriate place to stay upon arrival in the UK, or has stayed there previously.  So these cases are a little different from the unmarried partner visa.

The appeal process is typically long, and can be expensive if you require legal assistance.  If going through with an appeal, expect it to take around a year... so that's a year you're in visa limbo.  Also, if the visa was rightly refused (ex. due to lack of evidence), the appeal won't be useful as the original decision will be upheld.  An appeal is typically useful/successful if the applicant provided all the necessary documents/information with the original application, and something was overlooked by the entry clearance officer.  Your best bet if your visa is refused is to address the reason for refusal (ex. wait until you have a solid 2 years worth of evidence), and make a fresh application.  It might even be quicker that way.

I understand this isn't probably what you want to hear and I wish I could have better news, but the unmarried partner visa is notorious for the cohabitation evidence.  They are very picky about it.  Letters (that are not from official sources) aren't going to satisfy them.

ETA:  For clarity in some spots.

I'm not sure you are understanding...

You MUST have documentation in both names (individual is fine but you need twice as many)  at the same address covering the FULL 24 month period prior to application.  This is the fundamental requirement of the unmarried partner visa.  Without that exact evidence (the UKVI requirement) the application will be refused.  Period.  An appeal will fail.  While you could still eventually end up on a path to settlement in the UK, it will make each application complicated and cumbersome.  Why go through all that if you don't meet the requirements?

You only need to have received two items of mail at the parents...  If you don't have this, don't apply.

All the statements in the world are not going to be considered.  You must have the proof via mail.

Please really think about this.  I don't want you throwing good money after a bad application.  If you don't believe us, contact a solicitor.  Victoria Sharkey at Medivisas can advise you.