dependants

What is the name of your state? TX
If my father is on a fixed income such as s/s and does not make
over 10,000 a year and and I make make a little 15,000 could I
claim him as a dependant along with my son

In order to claim your father you must have provided over 1/2 of his support. If he received $10,000 frpm SS, and you only made $15,000 and supported yourself and your son the answer would be no you can not claim him.
Since your income is so low, it doesn't hurt too badly though that you can not claim your father. After your standard deduction and personal exemptions for yourself and your son, your tax liability is only around $200. Your child tax credit will wipe that away and if you qualify for the earned income credit, you will get roughly $2265 for that plus any federal taxes you had withheld on your W-2s.