A primary consideration will be the age and competence of your parent. If a parent is capable of making decisions they should be allowed and encouraged as much as possible. They need your support. Of course parents who are unable to make decisions will need you to make the decisions for them.
Sometimes with the best of intentions we rush in to help by making decisions for the survivor. Many parents will resent this infringement on their freedom, others may give in but become dependent or resentful. As a rule it is useful to offer advice and assistance while letting them know that the final decision is up to them. You may wish to offer to arrange funeral and memorial services, attend legal meetings, sorting and disposing of personal effects and helping with other practical matters. But it is not desirable to make funeral arrangements, legal decisions, sell property or dispose of personal effects without the full cooperation of a competent parent. While you have lost a parent, you need to remember that they have lost their life partner.
The death of a spouse will bring major life changes to your parent. Some decisions will have to be made immediately, but it is wise to advise your parent to avoid major life change decisions until time has been taken to permit reasonable decisions without regret. Grief can do strange things to our perspectives and decisions. In particular it is never wise to sell a house or move from an apartment when the pain of grief is intense. If a person thinks it would help them to grieve better or escape memories to be out of the home they shared with the deceased it might be wise to move to an apartment for several months first. After a reasonable trial a good decision can be made that will not cause unneeded financial hardship.
Your surviving parent was undoubtedly dependent on the deceased for many things. There will be a transition period where you can help the individual to learn many things needed for living without the deceased, such as cooking, taking care of finances, shopping and other practical matters.
It is important to remember that every person grieves differently. Recent studies have shown that the death of a spouse has profound, but very different effects on men and women. The New York Times has an excellent article which sums up the differences between how men and women grieve as well as more information on the subject. The same research has shown that men benefit greatly from men only support groups. The National Widowers Organization has a listing of resources specifically for men who have lost spouses.
There are many resources available online and within your community (and many online resources can connect you with local grief support groups). Both Widowed Village and WidowNet are good places to start your search for information on bereavement support for your parent.
Please share some ways you have found to help a parent cope with the death of a spouse.