Does the Parasyte story and/or manga continue after the anime? I've watched the Parasyte anime series but I haven't read the manga. The anime ends in a way that it could be the end of the story.

How is in the manga? Does the Parasyte story and/or manga continue after the anime?
According to Japanese Wikipedia, the Parasyte anime adapted the whole 64 chapters (10 volumes) of the manga, as such:


  
  stage:1  - chapter 1-2
  stage:2  - chapter 3-4
  stage:3  - chapter 5-7
  stage:4  - chapter 7-9
  stage:5  - chapter 10-12
  stage:6  - chapter 12-14
  stage:7  - chapter 14-17
  stage:8  - chapter 17-20
  stage:9  - chapter 20-22
  stage:10 - chapter 23-25
  stage:11 - chapter 26-28
  stage:12 - chapter 29-31
  stage:13 - chapter 32-34
  stage:14 - chapter 34-36
  stage:15 - chapter 37-39
  stage:16 - chapter 40-44
  stage:17 - chapter 44-47
  stage:18 - chapter 47-49
  stage:19 - chapter 50-52
  stage:20 - chapter 53-55
  stage:21 - chapter 55-57
  stage:22 - chapter 57-60
  stage:23 - chapter 61-62
  final stage - chapter 63-64
  


The storyline and the content are basically the same with most modifications related to cosmetic only (e.g. time period and character appearance).

However, there's a new spin-off manga called Parasyte Reversi which debuted in March 2018.