Do children follow parents' phobias genetically or by learning? Children's personalities come from  two sources: Parents' genes and the environment. 


Is there any research showing adults acquiring phobias as a result of being exposed to parents with the same phobia when they were a child?
For example if an adult is afraid of height, darkness or water, does this increase the probability of having parents with the same phobia?
Yes, phobias appear to be partly hereditary. This encompasses both genetic and environmental factors.

Kendler et al. (1999) review some work in this area:


  We have previously reported, from a population-based sample of female twins, that the liability
  to agoraphobia, social phobia and animal
  phobia was modestly infuenced by genetic
  factors with heritabilities estimated between 30
  and 40% (Kendler et al. 1992a). For situational
  and blood-injury phobia, by contrast, twin
  resemblance appeared to be due to familial
  environmental risk factors, which were responsible for ~30% of the variance in liability
  (Kendler et al. 1992a; Neale et al. 1994b).


A shortcoming of these studies is that they rely on single lifetime assessments, which may be unreliable. Kendler et al. (1999) tries to rectify this by following up with their original sample 8 years later. They conclude:


  Short-term reliability of the five phobias was modest (mean $\varkappa$=0.46), but higher than long-
  term stability (mean $\varkappa$=0.30). Unreliability occurred both for subject recall of unreasonable fears
  and for interviewer assessment of which fears constituted phobias. Examining fears and phobias
  together, in a multiple threshold model, results suggested that twin resemblance was due solely to
  genetic factors, with estimated total heritabilities, corrected for unreliability, of: any 43%,
  agoraphobia 67%, animal 47%, blood-injury 59%, situational 46% and social 51%.


They go on to compare their theory with other theories of phobia acquisition, such as classical conditioning and social learning theory. They state:


  By contrast, our results are supportive of an
  `inherited phobia-proneness' model (Seligman,
  1971; Gray, 1982)


This variance explained does, of course, leave considerable room for learned phobias. So, one should not overemphasize the hereditable components of fears and phobias (Delprato, 1980).

References


  Delprato, D. J. (1980). Hereditary determinants of fears and phobias: A critical review. Behavior Therapy, 11(1), 79-103.
  
  Gray, J. A. (1982). The Neuropsychology of Anxiety. Oxford
  University Press: New York.
  
  Kendler, K. S., Karkowski, L. M., &amp; Prescott, C. A. (1999). Fears and phobias: reliability and heritability. Psychological medicine, 29(3), 539-553. PDF
  
  Kendler, K. S., Neale, M. C., Kessler, R. C., Heath, A. C., &amp; Eaves, L. J. (1992). The genetic epidemiology of phobias in women: the interrelationship of agoraphobia, social phobia, situational phobia, and simple phobia. Archives of General Psychiatry, 49(4), 273.
  
  Neale, M. C., Walters, E. E., Eaves, L. J., Kessler, R. C., Heath, A. C., &amp; Kendler, K. S. (1994). Genetics of blood‐injury fears and phobias: A population‐based twin study. American journal of medical genetics, 54(4), 326-334. PDF
  
  Seligman, M. E. P. (1971). Phobias and preparedness. Behavior
  Therapy 2, 307-320