In the IRS's eyes, there isn't any such thing as an "unofficial employee". You're either doing it voluntarily as a family member, or you're an employee or a contractor.
First, you would need to determine if you are a W-2 employee or a 1099 contractor based on the facts and circumstances of your employment. Or, I suppose, if you're just doing it because you love your Dad and want to help out.
If a W-2, then your employer (maybe your Dad, maybe someone else in your family, depending on the situation) is required to issue you a W-2, pay their half of FICA taxes, and withhold and transmit your half of FICA wages to the federal government.
If a 1099 contractor, then your employer is required to issue you a 1099 if they make payments to you of over $600 in a year. You would be required to report that 1099 income, usually on a Schedule C. You would also generally be required to fill out Schedule SE if you made more than $400. You would pay both halves of the FICA taxes but would get an adjustment for 1/2 of that amount.
Your contribution to an IRA is limited to either box 1 wages (W-2) or net self-employment income (SE income minus business expenses minus 1/2 SE taxes).
Note that reimbursements wouldn't count as income. So if your Dad needed a walker, and you bought it for him, and he paid you back, that's not income to you.
Technically you're only required to file a tax return if your income is high enough or if you are required to pay SE taxes, plus a few other corner cases. So there can be cases where you're not required to file a return but can still have a little bit of income to qualify for an IRA contribution. Those circumstances usually involve a teenager babysitting for a few hundred dollars, as @disneysteve mentioned.
But even in those limited circumstances, you can still file a tax return to document the income, avoid failure-to-file penalties, start the statute of limitations on IRS audits, help prevent tax identity theft, and for your future records.
"At times the world can seem an unfriendly and sinister place, but believe us when we say there is much more good in it than bad. All you have to do is look hard enough, and what might seem to be a series of unfortunate events, may in fact be the first steps of a journey." Violet Baudelaire.