It depends who paid your support in 2020. If you had your own income you can file your taxes as independent  and claim your missing stimulus payments there; or your parents can amend they return and add you as a dependent. However you might not be eligible for some of the stimulus payments if you're a dependent. 

You cannot put anything about the 3rd stimulus on a tax return.   That check is not sent as part of a tax refund.  The IRS is issuing those checks separately on its own timetable.   The 1st and 2nd stimulus amounts are now based on 2020 returns---not 2019.    So your parents will not get any stimulus money by changing their 2019 return.   In fact --- if you were over the age of 17 they would not have received the 1st or 2nd amounts for you anyway.   Parents received stimulus money for dependent children who were under the age of 17 at the end of 2020.

If you cannot be claimed-----notice the word is CANNOT -- then you can get the stimulus amounts on your own 2020 return and later the IRS will likely send you a 3rd stimulus amount.   Or if you CAN be claimed then your parents will get the 3rd stimulus amount for you, since the age limit changed for the 3rd stimulus.   If your parents can still claim you as a dependent they can do so on their own 2020 return.

https://ttlc.intuit.com/community/dependents/help/when-do-i-have-to-answer-yes-to-being-claimed-as-a...

Do you fit the criteria for your parents to claim you as a qualifying child or qualifying relative?

WHO CAN I CLAIM AS A DEPENDENT?

You can claim a child, relative, friend, fiance (etc.) as a dependent on your 2020 taxes as long as they meet the following requirements:

Qualifying child

•They are related to you.

•They cannot be claimed as a dependent by someone else.

•They are a U.S. citizen, resident alien, national, or a Canadian or   Mexican resident.

•They are not filing a joint return with their spouse.

•They are under the age of 19 (or 24 for full-time students).

• No age limit for permanently and totally disabled children

  •      They live with you for more than half the year (exceptions apply).

Qualifying relative

•They don't have to be related to you (despite the name).

•They cannot be claimed as a dependent by someone else.

•They are a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.

•They are not filing a joint return with their spouse.

•They lived with you the entire year.

•They made less than $4300  (not counting Social Security)

•You provided more than half of their financial support. More info

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them.

Related Information:

•Does a dependent have to live with me?

•What does "financially support another person" mean?

•Can I claim a newborn baby?

Thank you so much for the response! I have one last question. Will my parents need to amend their 2019 tax return to get the missing 3rd stimulus or will they automatically receive/be able to claim it as missing on their 2020 tax return if they claim me this year (and they do not amend their 2019 tax return)? 

The third stimulus can be based on a 2019 return if a 2020 return has not yet been filed when the IRS decides to issue a 3rd stimulus check..  But amending a 2019 return will not yield a stimulus check for your parents.   It would take about six months to process an amended 2019 return.   If they file a 2020 return and are eligible to receive the stimulus based on their 2020 return, they will get it after they file the 2020 return.  And..again...there is not a place to request it, "claim" it,  or enter anything about it on the 2020 return.   It will be issued if their 2020 income qualifies them to receive it.