I was reading about Heidegger's concept of "Dasein" as being or existing in the moment. It struck a chord of similarity with the eastern concept of Zen. Is there a parallel?
No, Dasein does not mean 'being in the moment', but means
Dieses Seiende, das wir selbst je sind und das unter anderem die Seinsmöglichkeit des Fragens hat, fassen wir terminologisch als Dasein. (Sein und Zeit, §2)
(This entity which each of us is himself and which includes inquiring as one of the possibilities of its Being, we shall denote by the term "Dasein")
Simply put, it is Heidegger's formal name for the human being.
Zen on the other hand is the name of a form of Buddhism, or the meditation this form practises.
Paralells between Zen and Heidegger
However, there have been quite many studies and essays on the parallels between Heidegger and Zen. And Heidegger had some Japanese students who where Zen practitioners/scholars (e.g. Nishitani, Tanabe). For example, the present time for authentic Dasein is called 'Augenblick' (moment, instant), which some have interpreted as meaning 'being in the moment'.
Difference between Zen and Heidegger
The overall approach of Heidegger and Zen is nevertheless quite different. Heidegger wants to renew and (re)ask the question of Being. The essence of a human being is for him thinking. He focuses on philosophy and poetry.
Zen, so it seems to me (but I'm no expert), is about realizing the buddha nature, the emptiness of all, satori, etc. This realization is by letting go of thinking and it is usually not in/via language, but by meditation practice, koans.
So, both the goal and method of Heidegger and of Zen seem quite different to me.
I would like to submit this as an complement about Zen.
As Coinfold wrote at the comment line, but although, his citation , in which Taoism and Zen Buddhism is simultaneously analyzed is dangerous, in my opinion, the final summarization would be fine to me.
Western philosophers, it seems to me, favor to consider the "emptiness" in the eastern sense. If anyone has practiced Zen, then you eat, meditate, sleep regularly. If I take the "emptiness" literally, I may have to go more to Sokushinbutsu, in which "practitioners" cut meat, even cereals, and as is in the link,
The monks would slowly reduce then stop liquid intake, thus dehydrating the body and shrinking all organs.
To attain to the height or same being of Buddha.
Now, why was Zen practiced in the middle age Japan?
Because, they at the time, faced constant death. The then "sovereignty" Ashikaga Shogunate was very weak, battle was conducted yearly basis every where, ( please note, Ashikaga Shogunate was not able to issue its own currency due to the lack of power ), thus people felt "諸業無常" ( English translation ), in which they say worldly things are not permanent, changes the form constantly, ( even after the death ), and I think Zen's emptiness, if you would like to prefer to emphasize, is more close to that idea. And through or after realizing such emptiness, as Coinfold says, in a way very radically, Zen's final destination is the "transformation of the self". But I am afraid I can not agree Zen is "non-metaphysical". It's metaphysical in a sense that their aim is to become closer to closer to Satori, aka Buddha's final realization. To me, the most closest idea is that of Engels, who also saw worldly things changes constantly and do not possess fixed position ( but in Western sense ), and If I try to answer to the OP's question, the answer would be No.
I have frequently noticed a Buddhist influence in Heidegger's philosophy. A poignant case is Zen's focus on 'nothing', also aligning with aspects of negative theology, for example Meister Eckhard's philosophy, about which Heidegger writes
we find a remarkable parallel to the Hegelian determination of being and its identification with nothing. The mysticism of the Middle Ages or, more precisely, its mystical theology is not mystical in our sense and in the bad sense; rather, it can be conceived in a completely eminent sense.
Many commentators have remarked on the affinities between Heidegger's thought and East Asian traditions such as Vedanta, Mahayana Buddhism, and Taoism
There's a lot of google hits for Heidegger and Buddhism. Unfortunately, I have only read a monograph on Buddhism and Nietzsche, and material in general which merely mentions Heidegger in passing.
I would heisitantly say that it makes sense to say that Dasein is what experiences kensho, but then you may not find anyone who agrees with that.