State diagram for 2-bit parity generator
As given in the question, first we will try to describe the states and then will try to draw the state diagram.

The output of this machine depends on the present and previous input. So there can be four combinations:

-----------------------------------------
 Previous_input   Present_input   Output
-----------------------------------------
         0               0           0       S0 
         0               1           1       S1
         1               0           1       S2
         1               1           0       s3
-----------------------------------------


Since you need Moore implementation you have to consider 4 states corresponding to the four combinations given above. I have labelled the states using symbols S0-S3. From your sample input-output combination, it is clear that the initial state should be S0.

Part-a of your question can be answered from the above table.

Since we have got the states and corresponding output, the next step is to make the transition state table. The next state depends on the input and previous state. 

------------------------------------
 Present-state   Input   Next-state
------------------------------------
      S0          0         S0
      S0          1         S1
      S1          0         S2
      S1          1         S3
      S2          0         ??
      S2          1         ??
      S3          0         ??
      S3          1         ??
------------------------------------


I think you can replace the ??s in the table by your own. Drawing state diagram is simple once the this table is completed. Because, state diagram is the schematic representation of this table.