That is a really good question!! On a scientific level, I'd assume specific parts of the brain are involved with time perception, though don't know if they have been identified in birds - or in humans.
I do believe they have some awareness of time beyond preset routines. My job takes me away from home for 3 or 4 days per week. When I return, certain of my tame birds are more clingy than normal for a day or so. Don't notice this when back home after a few hours separation during the day. So they are definitely aware of very long absences, and have a specific response!!
It is surprising how little study has been done regarding this subject, especially where in the brain it is perceived.
Thinking of Time. Clearly there is a need to separate it into at least four groups:
Large Scale Time:
That which is connected to the movement of our Earth in conjunction with the Sun and Moon, which would also include the Stars - A Day! The location of the Sun and Moon as we perceive them to change location based on there the Rise and Setting points are each day - A Month and a Year cycle. As the individual ages and with setting of memories this scale grows to even longer (multi-year scale).
Short Scale Time:
The location of the Sun, Moon and Stars with in a given day - An Hour.
With in these first two groups is what allows long lived Creatures to set memories against; therefor, adjusting migration based on Sun cycles and Weather cycles. Remembering that Humans migrated for a very long time, even into modern times. In fact Human migration continues in North America everyyear with the year migration of the Snow Birds. Yes, a moment of light Humor, which is based in a truth!
Conceptual Time:
With the invention of devices that measured time. Time as understood by educated /knowledge exposed Humans changed. These devices allow (at first) a select group to predict with accuracy events of the past into the future. As the devices became more available the period 'DAY' changed to very precise measurements that Humans revolve around evermore. Today, most all of us are lost without a Time Device.
Large Scale Calendar Time:
This extends beyond the memory of an individual or individuals. At the point that Humans began setting down in a conveyable form multiple years, Large Scale Calendar Time came to be understood by Humans. As knowledge grew so did the ability of pushing the Scale not only back but forward in time.
So, to the question: Parrot's Perception of Time
In living with Amazon Parrots for well over 40 years, I have watched them use and understand both Short and Long Scale Time.
Clearly nearly all living things have a perception of the Time Period of a DAY! The living 'surface' creature adjusts their daily life to the DAY cycle. So yes, Parrots have an understanding of a Day /Night cycle and a general concept of whether it is early, mid or late Day /Night.
As a general statement, near all long lived /conceptual Creatures also have Large Scale Time understand as a result of aging and interaction with older members of the Social Group.
Answer: YES! but limited to the first two types!