Parachute activates on launch problem

Technically, you can escape Kerbin's gravity at any height, as long as you're going fast enough.

To orbit the planet at any given altitude, you only need to be travelling at the corresponding horizontal speed, as shown in the table below. The only reason that you can't orbit the planet consistently at low altitudes is that it's difficult to maintain your orbital velocity in the presence of atmospheric drag.

For example, to enter an orbit at 100km above the surface, you need a horizontal speed of 2245.8 meters per second.

Kerbin's atmosphere tapers off at approximately 68-70 km above the surface. Any orbit below this altitude will require corrections due to atmospheric drag, or else your ship will begin losing orbital speed and eventually deorbit. You can also do this intentionally if you want to deorbit your spacecraft - reduce your speed enough to drop below the minimum horizontal speed given in the table, and your spaceship will begin to descend.

I've snipped portions of this table to save space. The full thing can be found on the Kerbal Space Program Wiki:

Altitude (m)    Horizontal Speed (m/s)  Orbital Period (min)
35000                2357.9                   28.20
40000                2348.7                   28.54
45000                2339.6                   28.87
50000                2330.6                   29.21
55000                2321.7                   29.54
60000                2312.8                   29.88
65000                2304.1                   30.22
70000                2295.5                   30.56
75000                2287.0                   30.91
80000                2278.6                   31.25
85000                2270.3                   31.60
90000                2262.0                   31.94
95000                2253.9                   32.29
100000               2245.8                   32.64
110000               2229.9                   33.34
120000               2214.4                   34.05
130000               2199.2                   34.76
140000               2184.3                   35.48
150000               2169.6                   36.20
160000               2155.3                   36.93
170000               2141.3                   37.66
180000               2127.5                   38.39
190000               2114.0                   39.13
200000               2100.7                   39.88
300000               1980.6                   47.59
400000               1879.0                   55.73
500000               1791.5                   64.30
600000               1715.3                   73.26
700000               1648.0                   82.61
800000               1588.0                   92.32
900000               1534.2                   102.39
1000000              1485.5                   112.79
2,868,378            1008.9                   6 hours
8,140,000             635.4                   24 hours 

TLDR: Yes, you will lose chutes from version 1.0.1

Kerbal Space Program has an advanced physics engine and while the planets of the Kerbol system are different to the Sol system, the mechanics are similar:

In real life a re-entering spacecraft is hitting air at huge speeds and that is heating the vessel. The chutes undergo huge stresses when deployed - both from aerodynamic pressure and heating. If these stresses are too great, the parachutes could simply tear off.

Version 1.0.1 of KSP brings this real life danger into the Kerbal universe. Now, if you're moving too fast when the chutes deploy, they can be "burned up". Or, even if you're not undergoing heating, the chutes can be destroyed due to aerodynamic pressures.

How can I avoid chute loss?

Not losing parachutes from heating is easy: Don't deploy them when your vessel is burning up. You can tell if the vessel is burning up by looking at it - is it on fire? Don't deploy chutes.

Not losing parachutes from aerodynamic stresses is also easy: Don't deploy (even partially) when you're going fast. How fast is "fast"? I've conducted some tests and it looks like 1000m/s is the magic number. Although it may change depending on atmospheric conditions.

Edit: After version 1.1 chutes can be destroyed going slower than 1000m/s. Look at the chute icon in the staging, it changes colour based on whether it's safe to deploy or not.