Google is your friend in this case.
Obits appear in newspapers... SOME are online, some are not (and their records online could be just current ones).
You still can find the name of the biggest local newspaper, and email/ call them to find out what is the status of their archives.
1935 is way back.. that might involve the local library on microfilm. Old newspapers are also often kept at the state archives.
http://ssdi.rootsweb.ancestry.com/
this is functional for deaths in the US since about 1960ish.
It includes 3 women named Anna Tarpey, one died 1991 in New York. Unlike the other 2, it does not include the town/ zip code, but the birth on this one was 1907.
For ALL your research, you can join a county based group, or surname group, or even topical (ie Civil War). All the persons are zeroed in on one topic, so they tend to be more knowledgeable.
http://lists.rootsweb.ancestry.com/
http://www.oneidacountyhistory.org/
You may get help from someone here.
Wendy's tip for the day... what WORKS for one of your ancestors..will not be the right solution for the other one (even husband/ wife). Keep posting here. Unlike some of yahoo areas, people here are often happy to see several GOOD answers, more so than "competing" for points. There are many many sources that will come up as you work.. some online, some not. Each one is a learning key, that hones your imagination.