Is the following legally correct: my husband says that I have no parenting rights as a step-parent; and our home, which was purchased with my inheritance, is community property? 

As to your first issue, this isn't really a legal question; it's more of personal one.  This is something for you and your husband to work out amongst yourselves.  Step-parent or not, if this is your home and someone is living there for an extended period of time, then you should be well within your rights to ask for help with thing such as chores, etc.  Again, however this is a matter to be worked out between you and your spouse.

As to your second issue, typically any inheritance is deemed separate, not community property.  Even if the inheritance is used to purchase other property.  However, to the extent that your husband has contributed to maintaining/repairing the house, then he would be entitled to reimbursement and a share in the property's increased value do to such contribution. 

Note: Separate property can also become community property in several ways, including the failure to maintain the property as separate, or commingling with marital property, in a way that is too complex or undocumented to trace back to its source. 

For a fuller explanation of your situation you should consult with a divorce attorney in your area.