When you give align-items: center the text overflows the parent and the width is not constrained and so ellipsis will not appear.
If you remove align-items: center it works because align-items: stretch is the default - but the image get stretched out now:
.parent {
  border: 1px solid red;
  display: flex;
  /*align-items: center;*/
  flex-direction: column;
  width: 300px;
}
.text {
  white-space: nowrap;
  min-width: 0;
  overflow: hidden;
  text-overflow: ellipsis;
}
<div class="parent">
   <img src="http://placekitten.com/g/200/200" />
  <div class="text">
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum 
  </div>
</div>
 
 
Solution
Add width: 100% to the text - see demo below:
.parent {
  border: 1px solid red;
  display: flex;
  align-items: center;
  flex-direction: column;
  width: 300px;
}
.text {
  white-space: nowrap;
  min-width: 0;
  overflow: hidden;
  text-overflow: ellipsis;
  width: 100%;
}
<div class="parent">
   <img src="http://placekitten.com/g/200/200" />
  <div class="text">
    Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum 
  </div>
</div>
 
 
The min-width: 0; hack doesn't work when flex-direction is set to
  column.
Note that in min-width: 0 if for overriding the the flex item width as it is auto in the row direction (for column direction you use min-height). See example below for min-height: 0 used in a column flexbox: